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SURVEYING 



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THOMASON CIVIL ENGINEERINa COLLEGE, 



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PREFACE. 



This small work has Keen prepared primarily for the use of the 
Students of the Thomason Civil Engineering College, but it is 
hoped that it may be found useful to Overseers en Public 
Works generally in the Department Public Works, and others 
who are unable to purchase the more expensive works on the 
subject. With the above object, the compiler has endeavoured 
to make the work as complete as possible for all the more ordi- 
nary kinds of Surveying; and it is trusted that the insertion 
of certain minute details, (which have perhaps been omitted as 
unnecessary in larger works,) and the order in which the subjects 
have been placed — the chapter on each branch being complete 
in itself — may render the work particularly useful to begin- 
ners. 

It professes to be nothing but a compilation, the greater part 
having, by kind permission, been extracted from that valuable 
work, " Surveying for India," by Lieut.-Cols. Smyth and Thuillier. 
The Chapters on the Traverse Table, which it is believed are 
not to be found complete in any other book on Surveying, have 
been transferred in original. Erome's " Outlines of a Trigonome- 
trical Survey," Jackson's "Military Surveying," and Baker's 
" Land and Engineering Surveying," have also been made use 

of. 

I. P. W. 

ROOEKEE, 

March 2nd, 1865. 



INDEX. 



CHi-PTEE PAGE 

I. SUSVETII^^G- WITH THE ChAIIS- ONLY, 1 

Measuring Chains — Directions for using them — The Cross 
Staff and Offset Eod— The Perambulator— The Field- 
book — Example of a Chain Survey — Plotting the Survey. 

II. The Prismatic Compass, 13 

Description and Method of using the Prismatic Compass 
The Surveying Compass — Method of Surveying with 
the Prismatic or Surveying Compass — Plotting the Sur- 
vey — The Plane Table — Method of using it — Problem 
for finding your position from three known points. 

III. The Theodolite, 29 

The Vernier — Description of the Y Theodolite and its 
Adjustments — Everest's Double Arc Theodolite and its 
Adjustments — Method of observing with the Theodolite 
— Parallax of the Wires — Kepeating an Angle — Hints 
on the use of a Theodolite — Traversing with the Theo- 
dolite — Plotting the Traverse — Circular Protractor with 
Vernier. 

IV. The Teaveese System, 50 

Explanation of the mode of Surveying by Traverse — De- 
monstration of the proof of a Survey by Traverse — 
Amount of Error allowed, and method of distributing 
the corrections — Method of Plotting by Traverse — De- 
monstration of the Universal Theorem — The Traverse 
Table — Computation of the different Columns. 



Yl 

CHAPTER PAGE 

V, The Pocket Sextant, 80 

Description of the Pocket Sextant, and Method of adjust- 
ing it — The Artificial Horizon — Demonstration of the 
principal of the Construction of the Sextant — Angles 
taken by the Sextant not horizontal — Examples of find- 
ing inaccessible heights and distances by the Sextant — 
The Optical Square. 

VI. Levelling, . , , 90 

Definition of. Levelling — Curvature of the Earth — Re- 
fraction — Compound and Simple Levelling — Levelling 
Staves — The Y Level and its Adjustments — Gravatt's 
' or the Dumpy Level and its Adjustments — Troughton's 

Level and its Adjustments — Method to be pursued in 
li Levelling a tract of Country — Levelling with the Theo- 

I dolite — Contouring — Table of Corrections for Curvature 

j and Eefraction. 

i 

\ VII. Kailwat Curves, Useful Problems in Suevet- 

iNG, AND Scales, 119 

Simple Curves — Compound Curves — Serpentine Curves — 
I Curves of Deviation — Useful Problems in Surveying — 

Simple Scales — Diagonal Scales — Methods of reducing 

or enlarging Drawings — The Pentagvaph. 

i 

VIII. Teigonometeical Surveying, 150 

Selection and Measurement of a Base Line — Selection of 

Station — Observations of Angles — Reducing to the 

I Centre — Laying down the Triangulation on paper — 

Forms used in Field-book — Forms used in Calculation 
Book. 



CHAPTER I. 



THE MEASUEING CHAIN. 

Theee are two kinds of chains in general use for survey purposes ; 
one divided into feet, and the other into links. The former is 
generally a 50 or 100 feet chain, of which the first would only be 
used on account of its portabilifcy. The latter is 63 feet long, and 
is divided into 100 parts, each part being equal to a link, or 7*92 
inches ; it is called a G-aiiter's chain, and is specially adapted for 
measuring areas which are required to be computed in acres, roods, 
&c., for which it offers great facility, as its length is equal to four 
poles or perches, so that one square chain is equal to 16 square per- 
ches, or one-tenth of an acre. 

The length of a straight line must be found mechanically by the 
chain, and it is the most difficult operation in surveying. The sur- 
veyor, therefore, cannot be too careful in guardiug against, rec- 
tifying, or making allowances for every possible error, for, on 
the exactness of this measurement, the correctness of his work 
depends. 

The chain, however useful and necessary, is liable to many errors — 
first, in itself; secondly, in the method of using it ; and, thirdly, 
in the uncertainty of pitching the arrows. Every possible precau- 
tion must therefore be used. 

If the chain be stretched too tight, the rings will give, the arrows 
incline, and the measured line will be shorter than it reaUy is ; • on 

B 



2 



the other hand, if it be not drawn sufficiently tight, the measure 
obtained, will be too long. 

If the chain is a new one, it should, invariably, be measured 
daily until it lias stretcbed to its utmost ; if an old one, and which 
a surveyor will find by experience, to be always preferable, once in 
every three or four days is sufficient. A careful and correct sur- 
veyor, will, however, compare it daily, the mean of two compari- 
sons should then be taken as the length, for the work done in the 
interval. ^ 

Chains have been known to stretch as much as three inches in a 
day's work ; this, though trifling, in one chain, would be found of 
material consequence, after measuring 400 or 500 chains during the 
day, amounting, as such an error would, to nearly one chain and a 
half in the whole distance measured. 

The true length of a chain line measured with an incorrect chain, 
is easily found from the following proportion : — 

As the length of a correct chain is to the length of the chain 
used, so is the measured distance to the true one. 

For example, suppose the length of the chain to be J.00-32 feet 
or links, and the measured line 1050 feet or links, then, 100 : 10032 
: : 1050 : 1053-36, the true length of the line. 

To measure a chain, the ordinary offset rod, will, for general pur- 
poses, be sufficiently accurate ; two should be used in the following 
manner : — Stretch the chain pretty tightly on a level piece of ground, 
fixing two stout wooden pins at each end, in the handles of the 
chain ; then, lay down the two rods from one end, keep the second 
stationary, and, taking up the first, place it beyond the second, then 
keeping that stationary, take up the second and place it beyond the 
third, until the end of the chain be reached, when the decimals de- 
ficient, or in excess of the nearest foot, mav be measured with a 
Marquois, or other small, scale. 

It is a common practice to allow chainmen too much latitude in 
measuring lines, i. e., the surveyor is satisfied to come up at the end 



of the line measured, count the numher of links up to the station, 
depending entirely on the rear chainman for a correct account of 
the number of chains measured. This, even were the account of 
chains correct, (which is always doubtful,) can never be a satisfac- 
torily measured line. Unless the surveyor follows in rear of his 
chainmen, and keeps a continued watch on them, the probabilities 
are, that his work will have to be measured over again. 

A surveyor should accustom himself to follow his rear chainman, 
and satisfy himself as he is progressing, that he is measuring straight. 
To ensure the chainmen proceeding in as straight a line as possible, 
it is always well for the leading chainman to check the direction of 
the rear chainman, 'hj keeping the latter and the back station, (on 
which there is invariably a flag), in a straight line with himself. 
The 7*ear chainman does this, as he directs the leading one with the 
forward station, and thus, by a mutual check, great accuracy is 
obtained. 

Care must also be taken to see that the chainmen place the pins 
in the ground properly. The rear man should bring the handle of 
the chain up to the pin left in the ground, and the fore man, after 
getting in line, and stretching the chain, should put the pin in the 
ground, inside, and up against the end of the handle ; otherwise, if 
both pins be inside, or both outside of the handles, the thickness of 
the handle is gained, or that of the pin lost in each chain, which, 
when the distance measured is great, would amount to something 
considerable. If the ground is too hard to admit of the pin being 
driven in, a cross should be scratched on the ground, arid the piu 
laid down pointing to the intersection. 

Eleven arrows should be used instead of ten, as is generally the 
custom, for in the latter case, when the chain arrives at the end of 
the tenth arrow, thus denoting ten chains as measured, the chain is 
stopped!, and liable to be shifted ; whereas, with eleven arrows, one 
arrow always remains a fixture in the ground, and is never brought 
into the account, thus preventing the possibility of the chain being 



shifted whilst the other ten arrows are being taken to the leading 
chainman. 

Directions for using the Chain. — Flags are first to be set up at 
the places whose distances are to be obtained ; the place where the 
measurement is commenced may be called the first station, and that 
measured to, the second station. Two men hold the chain, one at 
each end. On the chain being stretched in the direction of the 
second station, the leader who is provided with eleven arrows, drives 
one firmlj into the ground, the rear chainman holding the other 
end at the first station ; he then proceeds in the direction of the 
seco7id station, until the rear chainman has arrived at the first arrow, 
when the latter directs the former in a liae with the first station, 
and a second arrow is firmly driven in, the rear chainman then takes 
up the fi.rst arrow, counts one chain as measured, and proceeds on 
until the eleven arrows are expended, one of which remaiuing in the 
ground, the other ten are sent on to the leading chainman. The 
exchange of the arrows is always notified by the rear chainman call- 
ing out with a loud voice, so many tens. The surveyor here marks 
in his Field-book that one change has been made, or 10 chains, 
or 1000 links measured. The chaiumen then proceed onwards, 
until another change has been made and entered, and so on, mark- 
ing every change until the second station be arrived at, when the 
number of arrows in the hand of the rear chainman will denote 
the number of chains, which, together with the odd links, and 
the number of changes that may have been made between the two 
stations, will make up the entire length of the line. 



THE CBOSS STAFF, AND OFFSET EOD. 

When the boundary of a survey has turns and bends in it, as is 
generally the case, it is not necessary to measure round every such 
turn and bend. The best and most usual way is, to proceed in a 




straight line from one principal corner to another, and when oppo- 
site to any bend in the boundarj, to measure the rectangular dis- 
tance, termed the ojfset, from the chain line to the bend, noting 
the same, together with the distance on the chain line from whence 
such offset was made. These offsets are geuerallv" measured with 
an offset staff or rod of tea feet. G-reat care is reijuired on the 
part of the surv^ejor in measuring offsets, for, unless the offset 
is taken at right angles with the chain line, the perpendicular 
measured for determining its area will be too long, and a correct 
result will not be obtained. 

A very convenient inslrument, called the " cross staff, " which can 
be made up bj any Bazaar Carpenter, is used for 
the purpose oi taking offsets. It consists of a 
piece of wocd, about six inches square and an inch 
and a half in thickness, fixed on the end of a staff 
about five feet in length, with an iron spike at the 
end, for the convenience ofplantingitinthe ground. 
The square piece on the top has two grooves ab and 
cd in it, about half an inch deep, at right angles 
with each other, made with a common saw. This 
instrument being placed any where on the chain 
line, if one groove be directed to the forward or 
back station, the other will of course give the per- 
pejidicular to the chain line. A well practised sur- 
veyor can, however, generally tell a right angle for an offset, with- 
out the assistance of this instrument. 

The best method of measuring offsets is, for the offset man to 
walk along the boundary, and to give a signal to the chain party, 
whenever he comes to a bend or corner ; the surveyor then places 
himself on the chain line in a rectangular position with the offset 
man, when the latter, measuring down towards him, gives in the 
length of the offset in rods, and returns immediately to the boun- 
dary to take up the next bend. A good offset man should never 



be taken off his work, for, by constant practice, he knows exactly 
when and where an offset is required. 



THE PEEAMBULATOB. 

This instrument is very useful for measuring roads, level plains, 
and everything where expedition is required. It does not give, 
however, a very correct measure in going over uneven surfaces, which 
is one of its principal objections ; and it is, therefore, only applica- 
ble to road and route surveys, where great accuracy is not essential. 

The following figure represents the English Pattern Perambulator, 
which consists of a wheel of wood A, shod or lined with iron to 
prevent the wear ; a short axis is fixed to this wheel, which com- 
municates motion by a long pinion fixed in one of the sides of the 
carriage B, to the wheel-work C, included in the box part of the 
instrument. For portability, the wheel A, is separable. 




In this iastrument the circumference of the wheel A, is eight 
feet three inches, or half a pole ; one revolution of this wlieel turns 
a single threaded worm once round ; the worm takes into a wheel of 
80 teeth, and turns it once round in 80 revolutions ; on the socket of 
the wheel is fixed an index, which makes one revolution in 40 poles, 
or one furlong ; on the axis of this worm is fixed another worm 
with a single thread, that takes into a wheel of 40 teeth ; on the 
axis of this wheel is another worm with 
a single thread, turning about a wheel 
of 160 teeth, whose socket carries an 
index that makes one revolution in 80 
furlongs or 10 miles. On the dial plate 
there are three graduated circles, the 
outermost is divided into 220 parts, or 
the yards in a farlong: the next into 
40 parts, the number of poles in a furlong ; the third into 80 
parts, the number of furlongs in ten miles, every mile being distin- 
guished by its proper Eoman figure. 




STJEVEFIiS-G BY THE CHAIN" ONLY. 

In making a survey with the chain only, we are confined to one, 
and the simplest geometrical figure, viz., the triangle, for of all 
plane geometrical figures, it is the only one of which the form can- 
not be altered, if the sides remain constant. That the triangle pos- 
sesses this property, is evident from the Theorem, (Euclid, vii. 1.,) 
which proves that " Upon the same base, and on the same side of 
it, there cannot be two triangles that have their sides, which are 
terminated at one extremity of the base, equal to one another, and 
likewise those which are terminated in the other extremity, equal 
to one another." 

The surface to be measured is therefore to be divided into a series 



i\ 



8 



of imaginary triangles ; and in this division it must be borne in mind 
that the triangles are to be as large, with reference to the whole 
surface to be measured, as is consistent with the nature of the ground ; 
for, by such an arrangement, we are acting on the important prin- 
ciple in all surveying operations, that it is well always to work from 
lolwle to part, and rarely from part to whole. 

The sides of these triangles are first measured, and as a neces- 
sary check on this part of the work, a straight line is in addition 
measured from one of the vertices to a poiut in or near the middle 
of the opposite side. This fourth line is called a tie-line, and is an 
efficient means of detecting errors, if any have been committed in 
the measurement of the sides of the triangle. This fourth measure- 
ment is made in accordance Avith a maxim which ought invariably 
to be acted upon in all surveying operations, viz., that where ac- 
curacy is aimed at, the dimension of the main lines, and the positions 
of the most important objects, should be ascertained or tested by at 
least two processes independent one of the other. AVithin the larger 
triangles, as many tie-lines and smaller triangles are to be measured 
as may be necessary to determine the position of all the objects 
embraced in the survey. The directions of the lines forming the 
sides of these secondary triangles are so selected or disposed that 
they shall connect and pass close by, as many objects as possible, 
so that the offsets to be measured from them may be as short and 
as few in number, as practicable. 

If the sides of these secondary triangles be in any case so distant 
from the objects whose positions are to be determined, as to require 
a length of offset greater then one or two chains, it then becomes 
advisable to construct, either on the whole or part of the side of 
the triangle as a base, a small offset triangle with the sides so dis- 
posed that they shall either embrace, or pass very near to the ob- 
jects to be measured by their intervention. 

The disposition and general combination of these triangles de- 
manding care and judgment, it is customary, previous to commene- 



9 

ing anj measurement, to walk over the ground for the purpose of 
obtaining a general knowledge of the surface, ajid of the relative 
positions of the most conspicuous objects. The acquisition of this 
knowledge depending on the coup d^oeil,is much assisted by an eye- 
sketch drawn with rapidity, and showing some of the principal 
roads, streams, temples, &c. 

This hand-sketch is not drawn to any scale, and its object is 
attained if it simply bear a general resemblance to a plan of the 
ground, as it will thereby assist the memory in the distribution of 
the surface into triangles. 

The sides of the larger ti'iangles are to pass as close as possible 
to the external boundaries to be surveyed ; the triangles should, 
moreover, be made to approach, as nearly as practicable, to the form 
of equilateral, avoiding with care very acute or very -obtuse angles, 
because the further the form of the triangle is removed from the 
equilateral, the greater will be the alteration in the form of the 
figure and its area, should any error have been committed in the 
measurement of any one of the sides. 

The triangles having thus been disposed to the greatest advantage, 
marks or pegs are placed in the grou.nd at each vertex of the tri- 
angles; the general form or position is then noted on the hand-sketch 
previously made, and distinctive letters are wiitten on the diagram 
at each point of intersection ; this arrangement admits of easy re- 
ference in the Field-book, or on the ground, to any triangle or part 
of a triangle. 

The points of intersection of all straight lines, as well as the 
vertices of the triangles, are always points measured to or fro^n ; 
they are called station points, and the lines connecting them, sta- 
tion lines, thereby distinguishing them from the simple offset lines. 
Stations are generally expressed by letters, main stations by capital 
or roman letters. A, B, C, &c., and secondary stations by small let- 
ters, a, h, c, &c. 

The hand-sketch, or rough diagram, is usually made in a Field- 

c 



10 

book, i. e., a book in which every minute step of the operations gone 
through, is to be entered with precision at the time. 

This Fie)d-book should be of a convenient size for the pocket, 
having the page ruled with a central column ; this central column 
is intended for all actual lines measured, and by commencing from 
the bottom of the page, the page becomes a smaller representation 
of the reality, with the line measured from you, and the offsets at 
their respective distances on that line, taken at so many links to the 
right or to the left, as they actually are on the ground and noted to 
the right or left of the central column. 

In keeping the Field-book, it first should ever be remembered 
that the central column is virtually but one line representing the 
Chain, the space within the column being merely required for the 
J I several distances on the Chain, whence the offsets are taken, and 

j|[ secondly, that all offsets read either way outward from the centre 

column, in the same way as they are measured outward from the 
Chain ; if the station line, therefore, should be crossed on the ground 
by a road or any boundary meeting it obliquely, its representation 
or type in the Field-book must not be made to pass obliquely across 
the middle column, but must arrive at one side of the column and 
leave it at the other, at points precisely opposite, as it would do 
were the middle column merely of the thickness of a line ; inatten- 
tion in this particular, causes much confusion in the relative position 
of offsets. 

To preserve uniformity, as it is more natural to measure from left 
to right, the place measured from is put on the left of the central 
column at the bottom of the line, and the station measured to is 
put at the top to the riglit ; the points of commencement and ter- 
mination of the line can thus be immediately seen. 

The book should be interleaved with blotting paper and the en- 
tries made in ink or inked in the same day on return from the field ; 
this should be rigidly adhered to, as any doubtful point is more 
likely to be remembered, or any error corrected, if the Field-book is 



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inked in at once and not left for two or three days ; the pages should 
also be numbered for facility of reference and each day's work dated. 
In taking offsets to corners of boundary marks or other objects, 
mark the relative position of the corner or object as to the chain 
line, and generally be careful to make the Field-book as much as 
possible ?ifac simile of the ground itself, with each boundary mark, 
&c., placed on the book, as to the central column, considered always 
as one line, in the same position as they stand to the chain on the 
ground ; no time is gained to the surveyor by hurrying over the 
notes in the Field-book, a little care in the Field saving mucli trou- 
ble in office. 

It cannot be too strongly impressed on the surveyor that the work 
which he is called upon to perform depends for its accuracy in a 
very great measure on the order, system, and neatness bestowed on 
all the steps whether of delineation or measurement : proper atten- 
tion in keeping the Field-book saves much time in plotting, and 
guards againsts the errors unavoidably arising from reference to a 
confused Field-book ; moreover, care bestowed in the first essays, 
will amply reward the surveyor, by giving accuracy of eye, freedom 
and steadiness of hand, qualities indispensible to his success. 

A specimen is here given of a Field-book, and the plan made from 
the Field-book, as an example of a chain survey. The line AB was 
first measured and offsets taken to the principal objects ; a and h 
were marked as points from which to run cross or tie-lines, but 
were afterwards found to be unnecessary. Next, the lines B C 
and C A, were measured, the points c in the former, and d, e in the 
latter, were noted, and then the cross lines, c d and h e were chained 
to enable ofi'sets to be taken to the objects in the middle of the 
triangle, and also to serve as check lines when plotting the survey ; 
these cross lines may be left nntil the sides of all the principle 
triangles are measured, and then each triangle may be filled in 
afterwards, but the points in the main line from which it is intended 
to run them must be marked at the time of measuring the sides of 

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the |. oipal triangles, or a great deal of unnecessary measure- 
ment will be entailed. Thus are all the sides of the triangles 
measured in succession, and their dimensions with the additional 
assistance of the offsets, give the means of ascertaining all boun- 
daries, external and internal, positions of houses, &c., and of finding 
the area of the whole and of every part, by direct computation 
from the Field-book. 

The method of plotting a chain survey is so self evident that a 
few words of explanation will suffice. In the above example, lay 
down on paper a line equal to A B, taken from any scale of equal 
parts, from the same scale take the length A C with a pair of com- 
passes, and with this as a radius and A as a centre, describe an arc ; 
now, taking E C as a radius and B 
as a centre, describe an arc, cutting 
the first, their intersection will give 
station C ; in the same way lay down 
the triangles C D B and C D E. 
Now, commence with A B, and mark 
off the distances given in the centre 
column of the Field-book, at the same 
time setting off the ofi"sets ; for this 
purpose a small cardboard or paper 
scale, of the shape shewn in diagram, 
will be found very useful ; by means 
of the middle scale either of the short 
arms can be placed at the required distance from the station and 
the offsets marked ofi-from them. In the same way proceed with 
the sides A C and B C, and then fill in the triangle by means of 
the cross lines c d and B^. Proceed with the other triangl 
similar mannar until the whole is completed 



[es m a 




y I lyuACE OF Kumc e\r g,oRE 




EXAMPLE 

A 

SURVEY 



EET TO AN INCH 

DO :oo eoo 700 boo spo loot 



CHAPTER II 



THE PEISMATIC COMPASS. 

The use of this little instrument is to measure horizontal angles 
only, or rather to take the bearings of objects, when the angle can 
be deduced from the two bearings ; and from its portability it is par- 
ticularly adapted for filling in the detail of a map, where all the 
principal points have been correctly fixed by means of the Theodo- 
lite. In the figure A re- 
presents the compass box, 
and B the card, which 
being attached to the 
magnetic needle, moves 
as it moves, round the 
agate centre, a, on v.hich 
it is suspended. The 
circumference of the card 
is usually divided to 15' 
of a degree, but it is 
doubtful whether an an- 
gle can be measured by 
it even to that degree of accuracy : c is a prism, which the observer 
looks through in observing with the instrument. The perpendicular 
thread of the sight- vane, E, and the divisions on the card appear 
together on looking through the prism, and the division with which 




14 

the thread coincides, when the needle is at rest, is the magnetic 
azimuth, or bearing, of whatever object the thread may bisect. The 
prism is mounted with a hinge-joint, D, by which it can be turned 
over to the side of the compass box, that being its position when put 
into the case. The sight-vane has a fine thread or horse hair stretch- 
ed along its opening, in the direction of its length, which is brought to 
bisect any object, by turning the box round horizontally ; the vane 
also turns upon a hinge-joint, and can be laid flat upon the box, for 
the convenience of carriage. F is a mirror, made to slide on or oil* 
the sight-vane E ; and it may be reversed at pleasure, that is, turned 
face downwards ; it can also be inclined at any angle, by means of 
its joint, d; and it will remain stationary on any part of the vane, 
by the friction of its slides. Its use is to reflect the image of an 
object to the eye of the observer when the object is much above or 
below the horizontal plane. AVhen the instrument is employed in 
observing the azimuth of the sun, a dark glass must be interposed, 
and the colored glasses represented at G- are intended for that pur- 
■pose ; the joint upon which they act allowing them to be turned 
down over the sloping side of the prism-box. 

At e is shown a spring which being pressed by the finger at the 
time ot observation, and then released, checks the vibrations of the 
card, and brings it more speedily to rest. A stop is likewise fixed 
at the other side of the box, by which the needle may be thrown 
off its centre ; "which should always be done when the instrument 
is not in use, as the constant playing of the needle would wear the 
point upon w^hich it is balanced, and upon the fineness of the point 
much of the accurac}^ of the instrument depends, A cover is adap- 
ted to the box, and the whole is packed in a leather case, which 
may be carried in the pocket without inconvenience. 

Prismatic compasses are now made with a silver graduated rim 
to the card, which is a vast improvement, and they should not be 
less than four inches in diameter. 

The method of using the instrument is very simple. First raise 



15 

the prism in its socket h, until you obtain distinct vision of the di- 
visions on the card, and standing at the place where the angles are 
to be taken, hold the instrument to the eye, and looking through 
the slit c, turn round till the thread in the sight-vane bisects one of 
the objects whose azimuth, or angular distance from any other ob- 
ject is required ; then by touching the spring e, bring the needle to 
rest, and the division on the card which coincides with the thread 
on the vane, will be the azimuth or bearing of the object from the 
North or South points of the magnetic meridian. Then turn to any 
other object and repeat the operation ; the difference between the 
bearing of this object and that of the former, will be the angular 
distance of the objects in question. Suppose the former bearing to 
be 40^" 30' and the latter 10° 15', both East or both West, from the 
North or South, the angle will be 30^ 15'. Prismatic compasses 
may be used with or without a stand. 

The divisions in some instruments are numbered from 0° to 180*^ 
South counting Eastward, and thence to 180° North counting "West- 
ward, others are numbered 6\ 10*^, 15°, &c., round the circle to 
330°, 90^ representing East, 180° South, 270^ West, and 360° 
North. These are by far the best, as the least liable to error 
in recording the results in a Eield-book, and are more generally 
understood by Natives. 

This instrument must be held or set up as nearly horizontal as 
possible, in order that the card may play freely ; also it must not 
be used near iron, or by a surveyor wearing steel spectacles. The 
variation of the needle must always be attended to, for if the fixed 
points above alluded to have been surveyed on the true meridian 
of the earth, the variation of the needle must be added to, or deduct- 
ed from, the observed bearing to obtain the true meridional bearing 
of the line. 



16 



THE SURTEYIKG COM T ASS. 

The Siirveyiug Compass consists of a compass-Lox, magnetic 
needle and two plain sights, perpendicular to the meridian line 
in the box, one of which has a longitudinal slit through which the 
surveyor lines the horse hair on the object of which the bearing is 
required; it is used for the same purpose as the Prismatic Com- 
pass, for filiiug in the interior detail of a survey by means of bear- 
ings. 

The sights are attached in various ways for portability, but are 
now generally made to turn 
down on a hinge, in order to 
lessen the bulk of the in- 
strument and render it more 
convenient for carriage, the 
diameter of the box varies 
from three and a half to 
four and five inches. "With- 
in the box is a graduated 
circle, the upper surface of 
which is divided into de- 
grees only, and numbered 
10°, 20^ 30 ^ &o., up to 
360'". The bottom of the 
box is divided into four 
parts or quadrants, each 
subdivided into 90^ num- 
bered from the jSTorth and 

South points each way, to the East and West points. The dis- 
advantage of a surveying compared with a prismatic compass is, 
that when looking through the sights, the angle is not presented 
to the eye, but it is necessary after fixing the sights on an object 
to read off the actual number of degrees pointed out by the 




17 



needle. It is evident from this that the surveying compass cannot 
be used without a stand, which is another disadvantage. This in- 
strument is however well adapted for first instruction of a Native, 
and is the same in principle as all the Vernacular Compasses made 
expressly for Native use. 



METHOD OF STJBVETINQ WITH THE PEISMATIC OR SURVEYING 

COMPASS. 

Let the annexed plan represent a survey of roads to be per- 
formed by the Prismatic or Surveying Compass. Having fixed on 




a starting point A, set the compass up there, and send a man with 

D 



18 

a, flagstaff to B, as far down the road as can be seen, and let it be 
placed in such a position that a long forward shot may be obtained 
the next time. Now, take the bearing of B, and proceed to chain 
from A to B, taking offsets to the sides of the road and any re- 
markable objects, precisely as in a chain survey. Having arrived 
at B the compass must be again set up, and a flagstaff having 
been sent to C, its bearing must be obtained ; then the line B C is 
to be measured, and so on. 

Angles must also be taken to any conspicuous objects that are 
out of reach of an offset, such as the corner of the house in the 
figure ; bearings from two points are sufficient to fix it, but a third 
should be taken as a check. The Pield-book is kept as in the chain 
survey, the only addition being the bearings. The angles to the next 
station are called the "forward bearings," the first of these should 
be written in the centre column immediately above station No. 1, 
and afterwards to the right or left of the centre column, according 
as the angle is to the right or left of the last direction ; this wdll pre- 
vent any mistake in plotting it. It is only necessary to take angles 
from every other station — for instance, we might go to B, and from 
there observe the beariugs of both B A and B C, then to D, and 
observe D C and D E ; but, it is advisable to take angles from every 
station, for it prevents confusion in the Field-book, and also the 
trouble of adding or subtracting ISO'* from every other angle, 
which in the other case would be necessary before the work could 
be plotted. 

For plotting a compass survey we require a protractor, which 
is an instrument for laying off angles. Protractors are either rect- 
angular, circular, or semicircular ; the former are most commonly 
used. They are made of either ivory, boxwood, or brass, about 
six inches long by two wide, and should be numbered in two rows, 
the outside one from 0° to 180°, and the inside from 180° to 
360°. 

To plot the above survey, having fixed on a convenient spot on 



19 



the paper for the stai'ting point A, (i. e.^ so that the survey may 
be contained in the paper, and as nearly in the middle as possible,) 
draw a line through it to represent the magnetic meridian, place the 




protractor to the right of this line, the edge coinciding with it, and 
the centre at the point A ; now, with a pencil, mark the required 
angle, and draw a line through this point and A, this will represent 
the first bearing ; on this line, produced if necessary, set off from 
any scale of equal parts, the length of AB, and through B draw the 
line NBS parallel to NAS, to represent another meridian, and plac- 

D 2 



20 



ing the protractor as before, lay off the angle NBC — set off BC the 
required length; and proceed with each angle in the same way until 
the end of the line EA should coincide with the starting point A. 
If we wish to measure an angle greater than 180°, the protractor 
must be placed to the left of the meridian, as at D, and the second 
row of figures used. Having completed the circuit and found it 
to be correct, w^e now proceed to lay off the offsets. 

The prismatic compass is very useful for what is called filling in 
a survey. The plotting of this kind of work is usually done in 
the field, each angle being laid down as soon as taken, and each 
distance and offset as soon as measured, so that no Field-book is 
required. A piece of paper with the work already done, such as the 
above circuit of road plotted on it, is placed in a sketching case, 
faint parallel lines having been ruled over it in the direction of the 
meridian, a quarter or Kalf an inch apart ; then, when an angle is 
taken, the protractor is placed parallel to these lines, (which can 
be done near enough for this kind of work by the eye,) and the 
angle is measured off. The following method of finding one's 
place in a survey with the prismatic compass will be found use- 
ful — referring to Mg. on page 17 — Suppose we wish to start the fil- 
ling in from a point Gr, and not from any of our former stations, the 
first thing is to find the position of the point G- on our paper ; to do 
this, we take the bearings of any two convenient stations, in this case 
D and E ; now, to find the bearing of our position y^'ow^ D and E, we 
have only to add 180° to, or substract it from, the bearings taken to D 
and E, and to protract the angles thus obtained from D andE — their 
intersection will be the point required. For instance, we find the 
bearings of D and E from Gr to be 100° and 205° respectively ; the 
former being less than 180°, we add that number to it, and the 
latter being greater, we substract it from it, and obtain 280° and 
15° respectively, placing a protractor at D and E, plot these 
angles, and their intersection gives the point G. It will be as 
\vell to take a bearincf to a third station as a check. Observe that 



21 



the nearer the two bearings meet at a right angle, the more 
accurate will be the station determined. 

If a large area had to be surveyed by the prismatic compass, it 
would be advisable to fix some points in it by triangles, starting 
from a measured base line ; these points should not be more than 
half a mile or a mile apart. Although they cannot be fixed very 
accurately when the angles are measured with an instrument which 
only reads to half degrees, yet with care they can be sufficiently so 
to serve as a check in the filling in, and oifer good points from 
which to start the circuits of traversing. 



THE PLANE TABLE. 

The Plane table used in India is of a very simple construction, 
it consists of an ordinary drawing board, varying from fifteen to 
twenty-four inches square, mounted on a tripod stand, and is move- 
able about an axis which goes through the head of the stand, and is 
fastened on the other side by a nut. Oak, teak, and toon, are the 
best woods for plane tables, deal and other soft woods imbibe 
moisture quickly and expand across the grain. 

A magnetic needle, in 
a compass box, is some- 
times attached to the 
table, it serves to point 
out the direction, and 
acts as a check upon the 
sights ; but, it mast be 
remembered, that it is 
never to be used inde- 
pendently. It is a dan- 
gerous thing in the hands 




2?. 



of natives, who are apt to set their table up by it alone, without 
ever verifying its position by sights. 

There is also an index or ruler, made of brass, iron, or wood, 
about the length of the diagonal of the table, at each end of this is 
a sight similar to those in the surveying compass ; one of the edges 
of the ruler is chamfered, and is called the fiducial edge ; the ver- 
tical hair and this edge are usually in the same plane, but this is 
not necessary. "When required for use, a sheet of paper is stretched 
on the board by first wetting it, and then glueing down the 
edges. 

To use the tahle. — Fix it at a convenient part of the ground, and 
make a point on the paper to represent that part of the ground. 

E;un a fine steel pin or needle throught this point into the table, 
against which you must apply tbe fiducial edge of the index, moving 
it round till you perceive some remarkable object, or mark set up 
for that purpose. Then draw a line from the station point, along 
this edge of the index. 

Now set the sights to another mark or object, and draw that 
station line, and so proceed till you have obtained as many 
angular lines as are necessary from this station. 

The next requisite, is the measure or distance from the station to 
as many objects as may be necessary by the chain, taking at the 
same time the offsets to the required corners or crooked parts of 
the edges, setting off all the measures upon their respective lines 
upon the table. 

Now remove the table to some other station, whose distance 
from the foregoing was previously measured ; then lay down the 
objects which appear from thence and continue these operations 
till your work is finished, measuring suck lines as are necessary, 
and determining as many as you can by intersecting lines of direc- 
tion, drawn from different stations. 

The use of* the instrument will be better understood from the 
following example, as given by Simms. 



23 



In the annexed diagram, let the points marked A, B, C, &c., be a 
few of an extensive series of stations, either fixed or temporary, the 
relative situations of which are required to be laid down npon the 
plan. Select two stations, as I and K, (considerably distant from 
each other,) as the extremities of a base line, from which the great- 




est number of objects are visible ; then, if the scale to which the 
plan is to be drawn is fixed, the distance, IK, must be accurately 
measured, and laid ofi" upon the board to the required scale ; other- 
wise a line may be assumed to represent that distance, and at some 
subsequent part of the work, the value of the scale thus assumed 
must be determined, by measuring a line for that purpose, and 
comparing the measurement with its length, as represented on the 
plan. 

Set up the instrument at one extremity of the base, suppose at 
I, and fix a needle in the table at the point on the paper represent- 
ing that station, and press the fiducial edge of the index gently 



PA 



agftiust the needle. Turn the table about until the meridian line 
of the compass-card coincides with the direction of the magnetic 
jieedle, and ia that position clamp the table firm. Then always 
keeping the fiducial edge of the index against the needle, direct the 
sights to the other station K, and by the side of the index draw a 
line upon the paper, to represent, the base IK ; when, if the scale is 
fixed, the exact length must be laid off, otherwise the point K may 
be assumed at pleasure on the line so drawn. 

But it is sometimes necessary to draw the base line first, when 
required, on some particular part of the board, so as to admit of the 
insertion of a greater portion of the survey ; when this is the case, 
the index must be laid along the line thus drawn, and the table 
moved till the further end of the base line is seen through both the 
sights ; then fix the table in that position, and observe what reading 
on the compass-card (or bearing) the needle points to, for the pur- 
pose of checking the future operations, and also for setting the 
table parallel to its first position, wherever it may afterwards be 
set up. It should be observed, that in placing it over any station, 
that spot on the table representing such station, and not the centre 
of the table, should be over the station on the ground : it may be 
so placed by dropping a plumb-line from the corresponding point 
on the underside of the table. 

Having fixed the instrument and drawn the base line, move the 
the index round the point I, as a centre, direct the sights to the 
station A, and keeping it there, draw the line IA along the fidu- 
cial edge of the index. Then direct in the same manner to B, and 
draw the line IB ; and so proceed with whatever objects are visi- 
ble from the station, drawing lines successively in the direction of 
C, D, E, &c., taking care that the table remains steady during the 
operation. 

This done, move the instrument to the station K, and placing the 
edge of the index along the line IK, turn the table about till the 
sights are directed to the station I, which if correctly done, the 



25 

compass-needle will point to the same bearing as it did at the for- 
mer station (in our example it was set to the meridian). Now, 
move the needle from I, and fix it in the point K ; lay the edge of 
the index against the needle, and direct the sights in succession to 
the points A, B, C, &c., drawing lines from the point K, in their 
several directions, and the intersection of these lines, with those 

drawn from the point I, will be their respective situations on the 
plan. 

To check the accuracy of the work, as well as for extending the 
survey beyond the limits of vision at I and K, the table may be set 
up at any one or more of the stations thus determined, as at E ; 
the needle being now fixed in the point E, on the board, and the 
edge of the index placed over E and I, (or K), the table may be 
moved round till the station, I, is seen through both the sights, and 
then clamped firm; the compass will now again (if all be correct), 
point outjts former bearing, and any lines drawn from E, in the di- 
rection of A, B, C, &c., in succession, will pass through the intersec- 
tion of the former lines, denoting the relative places of those objects 
on the board ; but, should this not be the case with all, or any of 
the lines, it is evident that some error must exist, which can be 
detected only by setting the instrument up, and performing similar 
operations at other stations. 

Having a number of objects laid down upon the plan, the situa- 
tion of any particular spot, as the bend of a road, &c., may at once 
be determined, by setting the instrument up at the place, and 
turning the table about till the compass has the same bearing as at 
any one of the stations. Clamp the table firm, and it will now be 
parallel to its former position, if no local attraction prevents the 
magnetic needle from assuming its natural position at the difi'erent 
stations. Eix a needle in the point representing one of the sta- 
tions, and resting the edge of the index against it, move the index 
till the station itself is seen through both the sights, and then draw a 
line on that part of the paper where the point is likely to fall. Ee- 

E 



26 

move the needle to another point or station on the board, and resting 
the index against it, direct the sights to the corresponding station on 
the ground, and draw a line along the edge of the index ; the point 
where this line intersects the last, will be the situation on the paper 
of the place of the observer. But, as a check upon the accuracy of 
the work, a third, or even a fourth line should be drawn in a similar 
manner in the direction of other fixed points, and they ought, also, 
to intersect in the same point. 

In this manner, the plane-table may be employed for filling in 
the details of a map ; setting it up at the most remarkable spots, 
and sketching by the eye what is not necessary should be more 
particularly determined, the paper will gradually become a repre- 
sentation of the country to be surveyed. 

The following problem, extracted from AVaugh's " Instructions 
for Topographical Surveying," may be found useful: — 

To fix a plane table in position at an unknown point x, by means 
of three points A,B,C, whose positions are laid down on the plane 
table, and represented by a,i,c, respectively. 

Pix a pin in the point h on the plane table, and placing the ruler 
against it and the point a, with the object and sight towards a 
{vide Fig. 1,) ; turn the table about, until the point A is intersected, 
then clamping the table in this position, turn the ruler and inter- 
sect the point Q, with the edge of the ruler still against the pin at 
I, and draw the line hn — now remove the pin to the point a, and 
unclamp the table — place the ruler agaiust the pin at a, and the 
point h, and turn about the table until the point B is intersect- 
ed, {vide Fig. 2,) ; clamp the table again and having intersected the 
point C as before, draw the line an through the intersection p of 
the line an and hm ; draAv the line cpq^ passing through the point 
c, and placing the edge of the ruler against this line, unclamp the 
table once more, and turn it about until the point C is intersected 
{vide Fig. 3,) ; now clamp the table, and it will be in position, and 
the unknown point co will be situated on the line cpq^ ; to find 



28 



tins point it is merely necessary to intersect either of the points as 
A, and draw the line Aax, and the accuracy of the operation is 
tested by intersecting the other point B and drawing the line 
JBlx which should intersect the line Aax^ on the line c^oq^ thus 
giving the position of x on this line. 

The demonstration of this problem is evident to those acquaint- 
ed with the same problem in Plane Trigonometry, it only remains 
to be remarked, that when the point C with regard to the point x is 
situated on the other side of the line AB, or below it, the lines 
an and hn will intersect on the opposite side of the line db, to 
that on which c is ; and if the point x be situated within the tri- 
angle ABC, these lines (an and hn) will diverge instead of con- 
verge, in which case they must be prolonged on the opposite direc- 
tion, until they intersect for the pointy. 

N.B. — The accuracy of the result depends on the length of the 
line cp. 



29 



CHAPTER III. 



THE YEENIEE. 

The Vernier is a contrivance for measuring oiF parts of the space 
between the equidistant divisions on the limb of a divided circle, 
arc, or any graduated scale ; it obtains this object by measuring 
the differences between the divisions of two approximating scales, 
one of which is fixed, and called the primary scale, the other 
moveable, and called, the vernier. 

If a number of parts equal to ^ — 1,* be taken from the primary 
scale, and a space equal to this be transferred to the moveable scale, 
and divided into n parts, these parts will each be smaller than the 
first by the n^^ part of a division on the primary scale. 

For let a = length of a division on the primary scale, 

h = length of a division on the moveable scale. 
Then by hypothesis 

{n — V) a =z nh ov 
71 a — a =. nh and 



a — - 
n 



a 7 

0. 



That is, h a division on the moveable, is smaller than a a division 
on the fixed scale, by the ;^^'^ part of a. This is the principle of the 
vernier. 



* Islote. — All this applies equally if w + 1 parts be taken, but it is more usual 
to take » — 1. 



30 

Suppose you have a theodolite, the horizontal circle of which is 
divided into 360°, and each degree into half degrees, or 30 minutes, 
and you want to construct a vernier to read to one minute. 

Y 

30 25 20 15 10 5 ^ 



TTT-J 



i'4. I's 1*2 A I'b -9'- «'• v'- el- si- *!• a'- ei- 



Take the length of 29 half-degrees on the horizontal circle, and 
divide that distance into 30 equal parts ; this forms the vernier, 
which is marked for convenience, 5, 10, 15, 20, 25, 30, from one end, 
or the zero. Set the vernier, which is so constructed as to slide 
evenly along the graduated limb of the instrument, to the horizon- 
tal circle, so that zero of the former may be in contact with zero of 
the circle ; then the last division, marked 30, of the vernier will, 
of course, agree with 14iJ®, or 29 half-degrees of the circle, and the 
proportion of each division of the vernier, will be to a division of 
the circle, as 29 to 30. If the zero of the vernier be moved from 
the zero of the circle, then the first coincidence that takes place 
between a division of the vernier with one on the circle, indicates 
the number of minutes passed over. To read off an angle on the 
horizontal circle, use a magnifying glass, and notice how many de- 
grees have been passed over by the zero of the vernier. Eor example : 
Let us suppose that the arrow at zero of the vernier has passed the 
21st degree of the circle : then, for the number of minutes in addi- 
tion, look along the vernier, until one of its divisions is found to 
agree exactly with a division on the circle below it : we will sup- 
pose, that the 14ith division of the vernier does so : then the angle 
is21M4'. 

I will give an example to show how to find the proper number of 
divisions for the vernier scale. Suppose the primary scale is divided 
to 10 minutes, and I wish to construct a vernier to read 10 seconds. 



31 

Then using the same letters as before, 

or 60a — 60d = a 
or 59 « = 60 5 

That is I must take 59 divisions from the primary scale, and 
divide them into 60 parts for the vernier. 



THE THEODOLITE. 

As an angular' instrument the Theodolite has from time to time 
received such improvements, that it may now be considered the 
most important one employed in surveying. They are of various 
modes of construction, but it will only be necessary here to de- 
scribe the two patterns in general use on the Indian surveys. 

Description of the Y Theodolite. This instrument (as represented 
in the next pagej , consists of two circular plates, A and B, called 
the horizontal limb, the upper, or vernier plate. A, turning freely 
upon the lower, both having a horizontal motion by means of the 
vertical axis, C. This axis consists of two parts, external and in- 
ternal, the former secured to the graduated limb, B, and the latter 
to the vernier plate, A. Their form is conical, nicely fitted and 
ground into each other, having an easy and a very steady motion ; 
the external centre also fits into a ball at D, and the parts are held 
together by a screw at the lower end of the internal axis. 

The diameter of the lower plate is greater than that of the upper 
one, and its edge is chamfered off and covered with silver, to receive 
the graduations : on opposite parts of the edge of the upper plate, 
or 180° apart, a short space, «, is also chamfered, forming with the 
edge of the lower plate a continued inclined plane : these spaces 
are, likewise, covered with silver, and form the verniers. The lower 
limb is usually graduated to 30 minutes of a degree, and it is sub- 
divided by the vernier to single minutes, which being read off by 



32 



the maguifjing glass, E, half or even quarter minutes can easily be 
estimated. 

The parallel plates, F tiud Gr, are held together by a ball and 
socket at D, and are set firm and parallel to each other by four 
milled-headed screws,* three of which, b, 5, i, are shown in the 
figure ; these turn in sockets fixed to the lower plate, while their 




* Nearly all Thaodolites are now made with three foot-screws, as hereafter 
described for the Everest's Theodolite. Thej^ possess great advantages over the old 
parallel plate screws, which can only be appreciated by a person who has used both. 



:33 

heads press against the under side of the upper plate, and being 
set in pairs, opposite each other, they act in contrary directions ; the 
instrument by this means is set up level for observation. 

Beneath the parallel plates is a female screw adapted to the staff 
head, which is connected by brass joints to three mahogany legs, so 
constracted, that when shut up, they form one round staff secured 
in that form for carriage, by rings put on them ; and when opened 
out they make a very firm stand, be the ground ever so uneven. 

The lower horizontal limb can be fixed in any position, by tight- 
ening the clamping screw, H, which causes the collar, c, to embrace 
the axis, C, and prevent its moving ; but, it being requisite that it 
should be fixed in some precise position more exactly than can be 
done by the hand alone, the whole instrument, when thus clamped, 
can be moved any small quantity by means of the tangent screw, I, 
which is attached to the upper parallel plate. In like manner, the 
upper or vernier plate can be fixed to the lower, in any position, by 
a clamp, which is also furnished with a tangent-screw. The motion 
of this limb, and of the vertical arc, hereafter to be described, is 
sometimes effected by a rack and pinion ; but this is greatly inferior 
where delicacy is required, to the slow motion produced by the 
clamp and tangent-screw. 

Upon the plane of the vernier plate, two spirit levels, d, d^ are 
placed at right angles to each other, with their proper adjusting 
screws : their use is to determine when the horizontal limb is set 
level : a compass also is placed at J. 

The frames, K and L, support the pivots of the horizontal axis of 
the vertical arc (or semicircle), ]M, on which the telescope is placed. 
The arm which bears the microscope, N, for reading the altitudes 
or depressions, measured by the semicircle, and denoted by the 
vernier, e, has a motion of several degrees between the bars of the 
frame, K, and can be moved before the face of the vernier for read- 
ing it off. Another arm clamps the opposite end of the horizontal 
axis by turning tlie screw, 0, and has a tangent-screw at P, by 

1-' 



34 



whicli the vertical arc and telescope are moved very small quantities 
up or down, to perfect the contact when an observation is made. 

One side of the vertical arc is inlaid with silver, and divided to 
single minutes by the help of its vernier ; and the other side shews 
the difference between the hjpothenuse and base of a right-angled 
triangle, or the number of links to be deducted from each chain's 
length, in measuring up or down an inclined plane, to reduce it to 
the horizontal measure. The level, which is shown under and 
parallel to the telescope, is attached to it at one end by a joint, 
and, at the other, by a capstan-headed screw, f, which, being raised 
or lowered, will set the level parallel to the optical axis of the teles- 
cope, or line of collimation ; the screw, g, at the opposite end, is to 
adjust it laterally, for trus parallelism in this respect. The teles- 
cope has two collars, or rings, of bell metal, groimd truly cylindrical, 
on which it rests in its supports, 7^, Ti, called Y's, from their resem- 
blance to that letter ; and it is confined in its place by the clips, ^, i, 
which may be opened by removing the pins, j, y, for the purpose 
of reversing the telescope, or allowing it a circular motion round its 
axis, during the adjustment. 

In the focus of the eye-glass are placed three lines, formed of 
spider's web, one horizontal, and two crossing it, so as to include a 
small angle between them ; a method of fixing the wires which is 
better than having one perpendicular wire, because an object at a dis- 
tance can be made to bisect the said small angle with more certainty 
that it can be bisected by a vertical wire. The screws adjusting 
the cross wires are shown at m : there are four of these screws, two 
of which are placed opposite each other, and at right angles to the 
other two, so that by easing one and tightening the opposite one of 
each pair, the intersection of the cross wires may be placed in ad- 
justment. 

The object-glass is thrust outwards by turning the milled-head, 
Q, on the side of the telescope, that being the means of adjusting 
it to shew an object distinctly. 



35 

A brass plummet and line are packed in the box with the The- 
odolite, to suspend from a hook under its centre, by which it can be 
placed exactly over the station from whence the observations are to 
be taken : likewise, if required, two extra eye-pieces for the teles- 
cope, to be used for astronomical observations ; the one inverts the 
object, and has a greater magnifying power, but having fewer glasses 
possesses more light ; the other is a diagonal eye-piece, which will 
be found extremely convenient when observing an object that has a 
considerable altitude, the observer avoiding the unpleasant and 
painful position he must assume, in order to look through the teles- 
cope when either of the other eye-pieces is applied. A small cap 
containing a dark-coloured glass is made to apply to the eye end of 
the telescope, to screen the eye of the observer from the intensity 
of the sun's rays, when that is the object under observation. A 
magnifying glass mounted in a horn frame, a screw driver, and a pin 
to turn the capstan-screws, for the adjustments, are also furnished 
with the instrument. 

The adjustments. 1st. TJw line of collimation, — The line of 
collimation in a Telescope is an imaginary line joining the centres 
of the object and eye-glasses ; it is evident, therefore, that the 
situation of this line holds a fixed relation to the tube and its 
appendages, so long as the object and eye-glasses maintain their 
fixity ; it is obvious then that, technically speaking, to speak of 
correcting the line of collimation is wrong ; what we ivisli to do is 
to bring the intersection of the cross wires into this line of colli- 
mation ; what we really do efi'ect, by^the following adjustment, is to 
bring the intersection of the cross wires into the line joining the 
centres of the Y's, which, if the instrument be properly made, 
should coincide with the line or axis of collimation. To do this, fix 
the intersection of the cross wires on some well defined distant 
object, turn the telescope half round in the Y's, and if the in- 
tersection of the cross wires still remains on the object, the instru- 
ment does not require adjustment in this respect ; but, if the 

Y 2 



•36 



horizontal wire be above or below the object, bring it back again 
to the point, half by means of the diaphragm screws, above and 
below the telescope, by first loosening one and then screwing up 
the other, and half by the tangent screw of the vertical arc, turn 
the telescope again half round in the Y's, and, if again off the 
object, repeat the operation till correct ; in the same way if on 
turning the telescope half round the vertical wire be to the right or 
left of the object, proceed in the same manner, moving half by the 
diaphragm screws at the side, and half by the tangent screw of the 
horizontal limb, and repeat till perfect. The intersection of the 
wires will now remain on the object during a complete revolution 
of the telescope in the Y's. 

Ind. The upper level. — The object of this adjustment is to set 
the bubble tube under the telescope parallel to the rectified line of 
collimation, (or what has been shewn to be the same thing, the line 
joining the centres of the Y's,) open the clips of the Y's, and 
bring the bubble to the centre of its run by the tangent screw of 
the vertical arc, now reverse the telescope in its Y's, turning it 
end for end, and if the bubble still remain in the centre, it is in ad- 
justment*; but, if not, bring it back to the centre, half by the 
adjusting screw at one end of the bubble, and half by the tangent 
screw of the vertical arc. E-everse the telescope again, and if still 
not correct, the operation must be repeated until it is so. 

^rd. Verticality of the axis. — The third adjustment is to set the 
bubbles attached to the horizontal plate, so that when in the centre 
of their run, they shall indicate the verticality of the axis, and, 
therefore, the horizontality of this plate. To do this having placed 

* The proof of this, viz : That on inverting the telescope, if the bubble re- 
main in the centre, it is parallel ; and that if it goes to one end or the other, it is 
not parallel to the line joining the centres of the Y's, is as follows: — suppose them 
to be parallel, then, since the bubble is in the centre of its ran the tube is horizon- 
tal ; and, therefore, the aforesaid line is horizontal. Now, on reversing, if the 
bubble is out of the centre, the tube is not horizontal, but the line joining the axis 
of the Y's has not been moved and is still horizontal ; therefore, our first supposi- 
tion that these two were parallel is not true ; but, on reversing, did the bubble 
again come to the centre, our supposition must be true. 



37 

the telescope parallel to two of the foot screws, bring the 'upper 
bubble to the centre of its ran, turn the instrument half round on 
its axis, and if the bubble be not still in the centre, bring it back 
half by the tangent screw of the vertical arc and half by the foot 
screws, repeat this until it remains in the centre both "ways, then 
turn the telescope a quarter round, bringing it over the third foot 
screw*, and by it bring the* bubble to the centre, the axis will now 
be vertical: to complete the adjustment, bring the bubbles attached 
to the horizontal plate to the centre of their runs by means of the 
capstan-headed screws at their ends. There only remains to deter- 
mine the zero of altitude ; when the axis has been made vertical 
and the telescope is horizontal, the vernier of the vertical arc should 
read zero, if it does not, the vernier plate may be unscrewed and 
placed so as to do so, but the better way is to note as an index 
error whatever the vernier may read and to apply it to all vertical 
angles. 

The other description of Theodolite used in India was designed 
by Sir G-eorge Everest, late of the Bengal Artillery and Surveyor 
Greneral of India ; it is known as " Everest's Pattern." There are 
two patterns, the double arc and the single are, both of which with 
their adjustments, will be here described. 

The accompanying diagram represents an instrument of the first 
kind. 

The horizontal circle, or limb, A, of this instrument consists of 
one plate only, which, as usual, is graduated at its circumference. 
The index is formed with four radiating bars, a, h, c, d, having ver- 
niers at the extremities of three of them, marked. A, B, and C, for 
reading the horizontal angles, and the fourth carries a clamp, e, to 
fasten the index to the edge of the horizontal limb, and a tangent- 
screw, f, for slow motion. These are connected with the upper 
works which carry the telescope, and turning upon the same centre 

* Or if an instrument with four foot screws, over the two remaining ones. 



38 



shdWany angle through which the telescope has been moved. The 
instrument has also the power of repeating the measurement of an 
angle ; for the horizontal limb being firmly fixed to a centre, move- 
able within the tripod support, E,, and governed by a clamp and 
tangent screw, s, can be moved with the same delicacy, and secured 
with as much firmness, as the index above it. 




The tripod support, which forms the stand of the instrument, has 
a foot- screw at each extremity of the arms which form the tripod ; 
the heads of the foot-screws are turned downwards, and have a 
flange (or shoulder) upon them, so that when they rest upon the 
triangular plate fixed upon the stafi'-head, another plate locks over 
the flange, and being acted upon by a spring, retains the whole in- 
strument firmly upon the top of the staff". The advantage of the 
tripod stand is, that it can easily be disengaged from the top of the 
staff", and placed upon a parapet or other support, in situations 
where the staff cannot be used. 

The telescope is mounted in the following manner.— The horizon- 
tal axis, L, and the telescope, M, form one piece, the axis crossing 
the telescope about its middle, and terminating at each extremity 



89 

in a cylindrical pivot. The pivots rest upon low supports, (only 
one of them, D, being visible in the figure,) carried out from the 
centre, on each side, by a flat horizontal bar, P, to which a spirit- 
level, Gr, is attached for adjusting the axis to the horizontal plane. 
The vertical angles are read off on two arcs of circles, H, H, which 
have the horizontal axis at their centre, and being attached to the 
telescope, move with it in a vertical plane. An index, upon the 
same centre, carries two verniers, I, I, and has a spirit-level, K, 
attached to it, by which it can be set in a horizontal position, 
so that whatever position the telescope, and consequently the 
graduated arcs, may have, when an observation is made, the mean 
of the two readings will denote the elevation or depression of the 
object observed, from the horizontal plane. On the upper part of 
the telescope is fixed a narrow box, containing a magnetic needle, 
for observing the bearings of objects. 

Adjustments. 1st. The loiver level. — Turn the telescope so 
that the lower level may be parallel to two foot screws, and by their 
motion bring the bubble to the centre of its run. If it remain so, 
on turning the telescope half round, the level is correct : but if not, 
the tube carrying the spirit glass is not perpendicular to the axis 
of motion, or, in other words, is not parallel to the horizontal plate. 

To remove this error, the bubble must be brought to the middle, 
half by the foot screws, and half by the adjusting screws at the ends 
of the level. Having perfected this, the levelling of the instrument 
is completed by turning the telescope a quarter round, so that 
one end of the level may be over the third foot screw, by which 
the bubble is to be brought to the centre of its run. 

2nd. The line of Collimation in Azimuth. — Having levelled the 
instrument as above, and having intersected some distant and well 
defined object with the cross hairs of the telescope, clamp the 
horizontal plates. Invert the telescope, and if the object be still 
intersected, the collimation is perfect. If not, correct half the 
error* in azimuth by the screws that give horizontal motion to the 



40 



diaphragm carrying the cross hairs, and the other half by the 
tangent screw, giving motion in azimuth to the instrument. Bepeafc 
till perfect. Two of the four screivs, ty icliicli the diapliragm is 
generally secured in the telescope, are only necessary therefore for 
this adjustment. 

Srd. The Zero of Altitude. — The above operations having been 
performed, bring the bubble of the upper level to the centre of its run, 
by the screws which retain the index, intersect an object and take 
the mean of the readings of the vertical arcs. Having inverted the 
telescope, and brought the bubble of the upper level to the middle 
of its run, obtain , in a similar manner, a second mean. Set the 
verniers of the vertical arcs so that the mean of the readiiigs shall 
equal the mean of the two means just found, and cause the cross 
hairs to intersect the object by means of the screws that retain the 
index, the bubble of the upper level, being thus thrown out, must 
be brought back to tlie centre of its run, by the adjusting screws 
at its extremities. E-epeat till perfect. 

The second kind differs from the Double Arc Theodolite, as 
its name implies, chiefly from having only a single arc ; on this 
account, vertical readings can only be taken on one face of the in- 
strument, for on inverting the telescope (by turning the axis end 
for end) and bringing the screws whicli retain the index under- 
neath, the index and the arc will be found to be on opposite sides 
of the axis. 

In other respe(;ts the instrument differs little from the double 
arc ; the single arc usually has only two, instead of three, verniers to 
read the horizontal plate ; and the level attached to the vertical 
index is fixed by the maker, and does not therefore admit of 
adjustment. 

The first two adjustments iu this instrument are similar to those 
for the double arc. 

^rd. The Zero of Altitude. — As before explained vertical read- 
ings cannot be taken with inverted positions of the telescope, it 



41 

therefore becomes necessary to use other means than those adopted 
with the Double Arc Everest, to fix the zero of altitude. 

Set up the instrument, and having levelled it, send a levelling 
staff with the vane set to the height of the axis of the telescope to 
any convenient distance, and having taken the elevation or depres- 
sion of the vane, change the places of the instrument and staff, 
level the former and set the vane of the latter to the present height 
of the telescope — again, take the reading of the vane. Now, the 
mean of this and the former reading will give the true inclination 
of one place to the other. Set the vernier to this mean, and by 
means of the screws which hold the index, bring the telescope to 
bear on the vane. This should complete the adjustment, but to 
see that the operation has been gone through correctly, it is ad- 
visable to repeat the process. 



THE METHOD OE 0BSEEYI:N"G WITH THE THEODOLITE. 

To level the Instrument. — The instrument being placed exactly 
over the station from whence the angles are to be taken, by means 
of the plumb-line suspended from its centre, it must fir^t be set 
approximately level by the legs of the stand, and then the levelling 
must be completed by the foot screws, Q, Q, Q, thus ; place the 
level, Gr, in a direction parallel to two of the foot screws, and by 
their motion, turning them both inwards, or both outwards, ac- 
cording as you want the bubble to go to the right or left, bring 
it to the centre of the tube : then turn the instrument a quarter 
round, so as to bring the bubble over the third foot-screw, and 
turn it to the right or left, until the bubble also becomes station- 
ary in the middle in this position. In performing this last opera- 
tion, the level, will be perhaps, thrown out on moving the instru- 
ment back to the first position, in Avhich case it must be again 
l-evelled. "When the bubble remains stationary in the middle 

G 



42 



during a whole revolution, the instrument is ready for obser\'a- 
tion. 

When you wish to intersect any ohjecfc, first move out the eye 
piece u.ntil you can see the cross wires distinctly, then by means 
of the focussing screw move the object-glass out until on moving 
the eye about, the image of the object ceases to move off the inter- 
section, of the wires and to have a fluttering, and undefined appear- 
ance. A picture of the field of view is formed in the telescope, and 
until it coincides with the plane of the wires this motion will take 
place ; just as when looking out of a window, the position of the bars 
with respect to the landscape change according as we change our 
position ; but if the landscape was painted on the glass (i. e., was in 
the same plane as the bars) this would not be the case. By moving 
the object-glass in and out, or as it is called focussing, we move the 
imaginary picture backwards and forwards in the telescope until 
it coincides with the plane of the wires, when there can be no mo- 
tion ; and the eye glass, which is merely a magnifier, having been set 
to see the wires distinctly, will also shew the picture so. The above 
motion is called par allajc, which may be defined as the apparent an- 
gular motion of an object arising from change of our point of view. 

To observe an Angle. — By means of the clamp, ^, and tangent- 
screw, y, (see last fig.) set the vernier marked. A, to 360° ; then 
turn the limb round, and with the lower clamp and tangent-screw, 
s, fix the cross-wires in the telescope on any object. Then loosen 
the upper clamp, e, and turn the upper limb round, fixing the cross- 
wires by the same clamp and tangent-screw on any other object ; 
the angle subtended can be then read 05" on the instrument. 

Another Iletliod. — Clamp the lower horizontal limb firmly in any 
position, and direct the telescope to one of the objects to be observed, 
moving it till the cross-wires and object coincide ; then clamp the 
upper limb, and by its tangent-screw make the intersection of the 
wires nicely bisect the object ; now read ofi" the two verniers, the 
degrees, minutes, and seconds of (either) one, which call A, and the 



43 



minutes and seconds only of the other, which call B, and take the 
mean of the readings, thus : 

A=142^ 36' 30* 

B= „ 37 



Mean=142 36 45 
'Next release the upper plate, and move it round until the telescope 
is directed to the second object, (whose angular distance from the 
first is required,) and clamping it, make the cross-wires bisect this 
object, as was done by the first ; again read off the two verniers, 
and the difference between their mean, and the mean of the first 
reading, will be the angle required. 

To repeat an Angle. — Leave the upper plate clamped to the lower, 
and release the clamp of the latter ; now move the whole instrument 
(bodily) round towards the first object, till the cross-wires are in 
contact with it ; then clamp the lower plate firm, and make the bi- 
section with the lower tangent-screw. Leaving it thus, release the 
upper plate, and turn the telescope towards the second object, and 
again bisect it by the clamp and slow motion of the upper plate. 
This will complete one repetition, and if read off, the difference 
between this, and the first reading will be double the real angle. It 
is, however, best to repeat an angle four or five times ; then the 
difference between the first and last readings (which are all that it 
is necessary to note) divided by the number of repetitions will be 
the angle required. 

The magnetic bearing of an object is taken, by simply reading 
the angle pointed out by the compass-needle, when the object is bi- 
sected : but it may be obtained a little more accurately by moving 
the upper plate (the lower one being clamped) till the needle reads 
zero, at the same time reading off the horizontal limb ; then turning 
the upper plate about, bisect the object and read again ; the difter- 
ence between this reading and the former will be the bearing re- 
quired. 

G 2 



44 



In taking angles of elevation or depression, it is scarcely necessary 
to add, that the object must be bisected bj the horizontal ^Yire, or 
rather by the intersection of the wires, and that, after observing 
the angle with the telescope in its natural position, it should be 
repeated with the telescope turned half round in its Y's, that is, 
with the level uppermost : the mean of the two measures will 
neutralize the effect of any error that may exist in the line of colli- 
mation. 

The altitude and azimuth of a celestial object may likewise be 
observed with the theodolite, the former being merely the elevation 
of the object taken upon the vertical arc, and the latter, its hori- 
zontal angular distance from the meridian. 

We here suggest a few hints on the use of these delicate instru- 
ments. 

1st. They must not be handled roughly. In taking them in and 
out of the box, it should be done with the greatest care, not knock- 
ing them against the sides of the box or forcing them into their 
positions within it ; the boxes are so constructed, that the instru- 
ment fits exactly into its own place, and unless it settles down of 
itself, forcing it will throw the instrument out of adjustment. 

2nd. J^ever permit a Native Surveyor to apply oil to any part of 
the instrument, under the idea that it will work easier ; a new in- 
strument will perhaps work stiff at first ^ but a very few days' use 
will rectify it, the application of oil is nothing but a restmg place 
for dust that is always flying about in the field : this dust works up 
into the various screws, wears them, and at the end of six months 
the instrument requires repair, or is next to useless ; if oil be neces- 
sary, it should be applied by the assistant, and then wiped off as 
dry as possible. 

3rd. Always throw the needle off its centre by the stop fixed on 
one side of the box, when the instrument is not in use, as the 
constant playing of the needle wears the pivot upon which it is 
balanced; and on the fineness of this point depends the accuracy of 



45 



the bearing. This is equally applicable to the Prismatic Compass 
and Circumferentor. 

4th. Always wipe the dust off the instrument on commencing 
and finishing a day's work, with a camel hair brush, as this will 
tend to prevent any accumulation of dirt about it : a Surveyor 
should partly be judged of by the state of his instruments, 

5th. "When once the variation of the needle is ascertained, never 
remove the box from off the telescope, for unless it be screwed on 
again, in the exact position it originally was, the variation of the 
needle will apparently alter. 

6th. On the care a Surveyor takes of his Theodolite, depends 
much of the accuracy of his work ; if he neglect and be careless 
about the former, he will one day have to lament over the accumu- 
lated errors of the latter. 

A survey of roads, &c., can be made with the Theodolite in 
the same way as that already described with the prismatic compass, 
and much more accurately, as the smallest Theodolites read to one 
minute of a degree, also another source of error is avoided, viz., that 
likely to occur from taking every angle with the magnetic meri- 
dian, the variation of which from the true meridian is not the same 
at different places, and in the same place at different times. la 
making such a survey wdth the Theodolite, called traversing, we 
should proceed precisely in the same manner as that previously laid 
down for the prismatic compass, the only difference being in the 
method of taking the angles which we will endeavour to explain. 

The Theodolite must be set up at the first station and levelled by 
means of the foot screws, the upper and lower horizontal plates must 
then be clamped at zero, and the whole instrument turned about 
until the magnetic needle steadily points to the NS line of the 
compass box, and then fixed in that position by tightening the 
clamping screw of the lower plate. 

iS'ow release the upper plate and direct the telescope to any 
objects that it may be advantageous to fix by intersection without 



46 

direct measurement to them, and having intersected them with the 
cross wires note the readings of the vernier in your Field-book, 
taking care to read from the same vernier as was before set to zero ; 
lastly, take the angle to your forward station where a staff* must be 
held for the purpose ; now leaving the horizontal plate clamped at 
that angle move your instrument to your second station, the dis- 
tance between the two must be measured and offsets taken. 

Arrived at the 2nd station, by means of the plumb-line place the 
theodolite over the mark where the staff was, and having levelled it, 
unclamp the bottom plate and with the vernier still at the last for- 
ward reading, turn the instrument bodily round and intersect the 
staff placed at the first station which is now the back station ; again 
tighten the clamp-screw and the instrument is now fixed in the same 
relative position as it was at the first station, and the upper plate 
may now be released to take angles to any conspicuous objects and 
to read the forward angle to station 3. To see that the above has 
been gone through correctly, after releasing the upper plate set the 
index to zero and the compass should, as in the first instance, coin- 
cide with the NS line of the compass box ; if not, an error has been 
committed in taking the last forward angle, or else the plate must 
have been moved from its position before the back station had been 
bisected : when this is the case it is necessary to return and exa- 
mine the work at the last station. If this be done every time the 
instrument is set up, a constant check is kept upon the progress of 
the work ; and this indeed is the most important use of the compass. 
Having thus proved the accuracy of the last forward angle, the an- 
gles from the 2nd station may be taken. At the 3rd, and every 
succeeding station, a similar operation must be performed, bisecting 
the back station with the instrument reading the last forward an- 
gle ; then the bearings taken to any conspicuous objects, and lastly 
the forward angle must be measured. After having fixed the teles- 

• A staff should be intersected as near the ground as possible, this prevents an 
error occurring if the staff be not placed perpendicular in the ground. 



47 

cope on the back station by clamping the lower plate, great care 
must be taken to prevent it from having the least motion whilst 
taking the other angle. Objects already fixed by intersections 
should continue to be observed to, so long as they are in view, as 
they serve as checks on the accuracy of the work. 

Prom this method of traversing, it will be easily seen that the 
angles taken at every other station, that is to say the 2nd, 4th, 
6th, &c., are 180^ out, and require that number to be added to or 
subtracted from them (according as they are less or greater than 
180°) before they can be plotted. The reason of this is obvious, at 
the 2nd station when we intersect the back station with the vernier 
at the last forward reading, the NS line of the compass box becomes 
parallel to the position that line occupied at the 1st station, but the 
N" now points where S did, and therefore the position of the theo- 
dolite is 180^ out. At the 3rd station it will be 180® out, from 
what it was at the second, and therefore in the same position as at 
the first. 

If the relative situations of some conspicuous points were pre- 
viously fixed by triangulation, there would be no necessity to have 
recourse to the magnetic meridian at all, as a line connecting 
the starting point with any visible fixed object, may be assumed as 
a working meridian ; the reading of the compass needle, should be 
noted at the first station when any such fixed object is bisected, the 
vernier of the horizontal plate reading zero, then at every succeed- 
ing station, upon the theodolite being set to zero, the compass 
needle should read the same as at the first. Indeed, even if no such 
points be fixed, it is better to use a line through your first station, 
and any conspicuous object as a working meridian, in preference 
to using the magnetic and if no object be available it would be 
better to lay down a referring mark. Eor, as before explained, the 
position of the North is continually varying, so that suppose you 
want to go over your work again or to start from the same place in 
a difierent direction, with the same meridian, having used the mag- 



48 

netic, you cannot be certain of doing so with any degree of cor- 
rectness. 

The method of plotting described for the compass survey is liable 
to some inaccuracies of practice, on account of having a new meri- 
dian for every particular angle to be laid down, and on account of 
laying off every new line from the point of termination of the pre- 
ceding one, whereby any little inaccuracy that may happen in 
laying down one line is communicated to the rest. As the angles 
taken by the theodolite are so much more accurate than those 
taken by the compass, it is as well to employ a more accurate means 
of laying them down, and this is gained to some extent ^by using the 
circular card protractor. This consists of a circle divided to quar- 
ter degrees, marked out on a square card, the centre part of which 
is cut out for the purpose of carrying on the protraction within it. 
The circle is numbered in two rows, like numbers being opposite 
to one another, i. e., the zero of the second row begins at the 180" of 
the first. The method of using this protractor is as follows : — A 
line having been taken to represent the meridian, place the protrac- 
tor on the paper with the zero and 180° points coinciding with it, 
and fix it in this position with weights or drawing pins to prevent 
its shifting, now apply a parallel ruler to the degree corresponding 
to the angle which it is wished to plot, one end coinciding with the 
number on the first row, and the other with the like number in the 
second row, on the opposite side of the circle, then slide the rule 
up to the point from which the angle is to be plotted and draw a 
line, which will represent the direction of the required angle. As 
soon as you have plotted to the extent permitted by the space cut 
out in the protractor, you must remove the protractor, draw a new 
meridian parallel to the first and apply the protractor to it as 
before. 

This method saves the trouble of shifting the protractor at every 
bearing, and also ensures greater accuracy in plotting, as a great 
number of bearings being laid down from one meridian, a trifling 



49 

error in the direction of one line does not affect tlie next ; the -accn- 
racy of the plot, howeyer, depends much upon using a ruler that 
moves truly parallel, which it is well to look to before using this 
method of plotting. A still more accurate method of plotting is by 
the traverse system, which will be explained in the next chapter. 



H 



50 



CHAPTER IV. 



THE TRAYEESE SYSTEM. 

A TEAYERSE may be defined as a circuitous route performed ou 
leaving any place on the earth's surface, by stages, iu different 
directions, and of various lengths, with a yIcav of arriving at any 
other place situated in any direction with reference to the former, 
and at any distance therefrom which cannot be reached in the 
direction of the shortest line connecting them. The angles which 
the stages or station lines form with the meridian are called '' bear- 
ings" the quantity of Northing or Southing made in each distance , 
is called the difference of latitude, and the amount of Easting or 
Westing is termed the dej^arture, 

. When the bearing corresponds with the meridian, or with the 
perpendicular to it, there will in the former case be nothing but 
difference of latitude, and in the latter nothing but difference of 
departure, and the distance measured will itself, express the amount 
of Northing or Southing, or of Easting or Westing due to the 
change of position. The perpendicular to the meridian coincides, 
at first, with the small circle of latitude. When the distances are 
great, the deviation of these two becomes sensible, being the differ- 
ence between the base and hypothenuse of a right-angled spheric 
triangle. In ordinary survey work the difference is scarcely 
sensible. 

When, hoAvever, the bearing does not correspond with the meri- 
dian or with the perpendicular to it, there will be for every dis- 
tance measured a certain corresponding change both in latitude 



51 



and longitude (or departure) ; and as these will, with reference to 
their particular distance, answer the condition of our definition, 
they may, with propriety, be termed the traverses of the distances : 

"We will therefore define : 

1st. Meridians as North and South lines, which are supposed to 
pass through every station of a survey, running parallel to each 
other.* 

2nd. The difference of latitude or the Northing or Southing of 
any line, as the distance that one end of a line is North or South 
from the other end. 

3rd. The departure of any line, as the perpendicular distance from 
one end of the line to a meridian passing through the other end. 

It is proved in Euclid I. 32. Cor. I. " that all the interior angles 
of any rectilineal figure, together with four right angles, are equal 
to twice as many right angles as the figure has sides," or, in other 
words, that — 

In any rectilineal Jigure, the sum of all the interior angles, is 
equal to twice as many right angles as tlie figure has sides, less four 
right angles. 

The Traverse System is a method of computation by rectangular 
co-ordinates, and is applicable to any mode of surveying whatever 
such as Honte Surveys, Kail way Lines, Navigation Courses and the 
like, where every station is fixed by the distances on the meridian 
and perpendicular, and this is essential to Gale's System, which may 
be termed the periphery measuring or perimetrical method. By 
throwing a series of angles over the face of a country, and forming 
a network of large circuits, the liability to error is reduced witliin 
the narrowest limits, which the means at disposal permit. This 
angular Circuit System, in extensiye operations in a country like 



* Tliese meridians are not really parallel, but converge towards the poles of tlie 
earth, but so insensibly as not to be worthy the notice of a Surveyor's opt-rations 
within a very linnited space. In extended operations, however, as in India, w'herc 
whole provinces come within the Traverse System of the Revenue Survey, the paral- 
lelism of the meridians must be preserved, to carry out in practice the accuracy of 
the above Thcoiem. 

fi 2 



52 



India, with instruments of the best construction and moderate 
power and size, can alone enahle a surveyor to carry out in prac- 
tice the theoretical accuracy of the Traverse, and permit, by the 
aid of logarithmic calculation, an approximation to the proof 
req[uired, viz : — 

1st. "That the sums of all the interior angles shall be equal to 
twice as many right angles as the figure has sides, less four right 
angles," and — 

2nd. As regards the linear measurements, " That the sums of the 
Northings be equal to the sums of the Southings, and the sums of 
the Eastings be equal to the sums of the Westings ; which latter 
wiU be presently explained. 

However correctly distances may be measured, unless the angular 




work is also correct, the result will be unsatisfactory, but with 
both these data accurately determined, the proof will be certain, 



53 



and it will be observed, how admirably each step in the work 
proves the other, and what confidence the system gives to a sur- 
veyor who has no need whatever to put any of his work on paper, 
but with his Traverse correct, may produce his map at any future 
period with undoubted certainty. 

"We will now proceed to explain the mode of Surveying by 
Traverse. 

Draw any figure such as ABCDEFGrHIJA, representing the 
sides of an irregular Polygon. 

If the Theodolite is first set up at the station A, and the interior 
angle JAB is observed, and then at B, observing the interior angle 
ABC, at C, the interior angle BCD, and so on all round the poly- 
gon, then will the sum of all the interior angles, JAB, ABC, BCD, 
&c., be equal to ten times two right angles (the figure having ten 
sides) less four right angles or 180^ X 10 — 360^ — 1440°. 

In practice it will be found that this result caunot be exactly 
attained, and that the sum of the angles will generally amount tO' 
two or three minutes more or less ; to meet this, a correction of 
one minute in every four or five angles, additive or substractive as 
the case may need, is generally necessary to obtain the result re- 
quired. 

Having thus proved the angular work correct, the next operation 
is to obtain the bearings of the several sides of the polygon or 
angles subtended by these sides with the meridian. This is either 
done by the magnetic needle on the theodolite or by astronomical 
observation, but as all should progress on the true meridian of the 
earth, we shall therefore treat only of true meridional bearings or 
angles formed by each line with the true meridian. If the Theodo- 
lite were adjusted in the plane af the meridian on every station of a 
survey, w^e should find no difficulty in obtaining the true bearing of 
each line, but as this would be very troublesome and next to impos- 
sible, it is only necessary in practice to obtain the correct bearing 



5J. 

of the first line of a survey, from which by the assistance of the an- 
gular work the bearings of the other lines can be deduced. 

This true bearing being once established has only to be checked 
and corrected by similar means after every 50 or 100 square miles 
of country traversed, and it will be found seldom to exceed from 5 
to 10 minutes of a degree from the true meridian, Avhereas, if the 
magnetic needle was used, an error of 15 or 30 minutes is scarcely 
traceable in a single ol:)servation, and where so many instruments 
are in use, all giving a different magnetic variation, it is plain that 
without this method of deducing the azimuths from the angular 
observations, the utmost confusion would arise. 

Bide. — To the bearing of the line preceding that of which the 
bearing is sought, add the inward angle formed by these two lines, 
and the sum increased or diminished by ISO"", according as it may 
be less than, or in excess of 180^, will be the bearing of the next 
line sought. 

Before proceeding to prove the aboA^e rule, we will first premise 
that, in all modern theodolites the divisions are numbered round 
the circle, from 0'' to 360°, so that the bearing of any object 
between 0° and 90° is reckoned North-East, between 90' and 180^ 
South-East, between 180° and 270^ South- West, and between 270° 
and 360° North- West, 0°, 90°, 1S0°, 270^, being respectively due 
North, South, East, and West. 

This method of reckoning the bearings of objects is by far the 
most convenient for practice, as without the necessity of making 
iise of the letters to denote the direction, the bearing is known at 
once by the number of degrees contained in the arc. 

Let the bearing of the line AB in the following figure be given, 
as found by astronomical observation. 

Ih find the learing of tlic line BC. — Produce AB to a, and CB 
to c. 

The two meridians NS, and N^S' being parallel, the angle WBa 
is e(iual to the angle NAB, if to the anolc ^'B« or arc NV^ wc add 



5^ 



the interior angle of the polygon ABC, or its equivalent in arc 
aS^c, we obtain the angle formed by the line CB with the meridian 




N'S^ or arc N^«S'c, if then the angle cBC or 180° be deducted from 
this, thus reversing the direction of the line, we have left the angle 
!N^BC or bearing of the line BC with the meridian ]N'^S\ 

To find the hearing of the line CD. — Produce BC to 6, and 
DC to d. 

The two meridians N^S' and N^S^ being parallel, the angles IS'^BC 
and N^C6 are equal. If to the angle Is'^Cb or arc Wh, we add the 
interior angle of the polygon BCD, or its equivalent in arc ftS'J, 
we obtain the angle formed by the line DC with the meridian IS'^S^ 
or arc Wh^-d, if tlien the angle dCJ) or 180° be deducted from 



5G 

this, thus reversing the direction of the line, we have left the angle 
WCD or bearing of the line CD with the meridian WB\ 

To find the hearinj of the line BE. — Produce CD to e, and 
ED to/. 

The two meridians N'S' and N^S^ being parallel, the angles 
N^CD and WJ)e are equal. If to the angle WDe or arc N^d we 
add the interior angle of the xjolygon CDE, or its equivalent in arc 
ffi we obtain the angle formed by the line ED with the meridian 
N'S=5 or arc ^^efi if then the angle /DE or 180° be added to this, 
thus reversing the direction of the line, we obtain the angle N^DE 
or bearing of the line DE with the meridian N^S^ 

To find the leaving of the line EF. — Produce DE to y, and 
FE to h. 

The two meridians N^S^ and N^S* being parallel, the angles, 
N^DE and WE^g are equal. If to the angle WEg or arc NViS*^" we 
add the interior angle of the polygon DEE or its equivalent in arc 
gl^^h, we obtain the angle formed by the line EE, with the meridian 
JST^S^ or arc N*AS*^T^Vi from which if we deduct the angle 7iEF or 
180^, thus reversing the direction of the line we have left, the 
ungle N*EE or bearing of the line EE with the meridian N*S*. 

And so on, this proof may be carried through every line of the 
polygon, until we come to the last line JA, when its bearing added 
to the interior angle JAB -J- or — 180*, as the case may require, 
will give the original starting bearing of the line AB. 

"We have been thus prolix in explaining how the bearings of 
the above four lines of the polygon are obtained, as they contain 
cases in each quadrant of the circle, BO, being a South-East bear- 
ing, CD, North-East, DE, North- AVest, and EF, South- West ; the 
same rule is however applicable to the remaining sides.* 



* If tlie sum of the preceding bearing and forward angle after deducting 180° 
amounts to more than 360", deduct 360° from the total, the remainder will be the 
bearing of the next line. 



57 



THE PROOF OF THE TEA.VERSE, AMOITI^T OF EREOR ALLOWED, 
A:XD METHOD OF CORRECTIOI^. 

We now offer for consideration the following Theorem, viz. : 

That in every Survey, correctly taken y the sum of the distance gone 
North from a certain ^oint, will he equal to the sum of the distances 
returned South to the same point, and that the sum of the distances 
gone East, will he equal to the sum of the distances returned West. 

The truth of the above is self-evident, for the meridians within 
the limits of an ordinary Survey having no sensible difference from 
parallelism, it muBt necessarily follow, that if a person travel any 
way soever within such limits, and at length come round to the 
place where he set out, he must have travelled as far to the ISTorth 
as to the South, and to the East as to the "West, though the prac- 
tical surveyor will always find it difficult to make his work close 
with this perfect degree of exactness. 

We will, however, explain this more fully with the assistance of 
a diagram. 

Let the line NS run due North and South, and EAV due East 
and West. If we fix on the point A, as a starting point, and a 
person walks from A to h, on the line NS, say 400 yards, and 
wishes to return to A, he must walk back 400 yards ; in going 
therefore from A to h, and back from h to A, he has walked 400 
yards North and returned 400 yards South. In the same manner 
if he fixes on the point 5 as a starting point, and walks to 13 on the 
line EW, say 300 yards to get back to h, he must return 300 yards ; 
in walking therefore to B, he has gone 300 yards East, and returned 
300 yards West. 

Supposing now, he walks from A to B, say 500 yards in the direc- 
tion of the line AB, he will then have gone North from A, 400 
yards, and East from A, 300 yards. 

In a continuation of the figure, having walked or measured from 
A to B, he proceeds on and measures from B to C, in doing so, he 

I 



58 



goes a certain distance Soutb and a certain distance East of B to 
arrive at C, thence he measures to D, going a ceriain distance 




north and east of C to arrive at D, from D he measures to E, from 
E to E, and so on, going north, south, east, or west, from the pre- 
ceding station as the direction of the line may he, until he arrives 
back at his original starting point A. In making this tour, there- 
fore, he has gone the same distance north as he has returned south, 
and the same distance east as he has returned west. 

Let the vertical and horizontal lines drawn through the several 
stations A, B, C, &c., represent, the former, a series of meridians 
or north and south lines or lines of longitude; and the latter, a 
series of east and west lines or lines of latitude : as these lines of 
latitude and longtitude are all respectively perpendicular and jmral- 



59 

iel to each other, it follows that the angle formed by the intersec- 
tion of the meridian lino of one station, and the latitudinal line 
of the next station as at 6, k^ Z, m^ &c., must be a. right angle 
or 90^. 

JSTow supposing all the lines AB, BC, &c., to have been carefully 
measured with a chain, and that haying obtained the bearing of 
the line AB, by astronomical observation, we have deduced the 
bearings of all the other lines by the rule (page 53) ; we then 
have the data in each line, of a side and two angles to find the 
other two sides. 

Eor instance, in the triangle ASB, we have the side AB, and the 
two angles 5AB, A6B, (the latter being invariably 90° or a right 
angle,) to find the other two sides Ah and 6B, the former being 
the difierence of latitude, and the latter the departure of the station 
B from A. In like manner, in the triangle B/tC, we have the side 
BC, and the two angles CB/c (obtained by deducting NBC from 
180°) and 'EkQ (a right angle), to find the other two sides BA; and 
Cky the former being again the difference of latitude, and the latter 
the departure of the station C from B, and so on for every line 
round the figure. 

The object of calculating all the sides of these several right- 
angled triangles on each line, is to obtain the difference of latitude 
and departure of each station from the preceding one, which 
difference being found, the sums of all differences of latitude of 
lines going North, must equal the sums of all differencea of 
latitude of lines going South, or 

A6 + CZ + Dwi + G^ + J5 = Bh + E;^ + Fy + H^^ + Ir 
and the sums of all differences of departure of lines going East 
must equal the sums of all differences of departure of lines going 
"West, or 

IB 4- kC + ZD + rZ 4- sA = wE + ??F + yG +^H + rjl 
and if this is not the result of the above calculations, the Survev 
has net been truly taken. 



GO 



We have before stated, that in the measurement of angles, a 
certain correction is allowed in practice, to obtain the result of the 
Theorem which forms the basis of the work, so also in the 
measurement of Chain lines, a correction is necessary to meet 
the errors, that notwithstanding the greatest care, will occur. In 
actual practice, the columns of latitude and departure will not 
balance exactly, for inaccuracies must arise from observations and 
chaining in the field, which no care could obviate. To adjust these 
differences, previous to defining the meridian distances, the rule is, 
that should the discrepancy amount to one-fifth of a Pole or five 
Links for every station, it will be clear an error has been made in 
the field measurements, which must be discovered by a re-survey. 
When difierences, however, are within these limits, the amount of 
error allowed is one link in ten chains, additive or subtractive 
from the sums of the Northings and Southings to correct the lati- 
tude, and from the sums of the Eastings and Westings to correct 
the departure. 

This error must be apportioned among each of the distances of 
the Survey by the following proportions, viz. : 

As the sum of all the distances is to the whole error, so is each 
distance to its correction. 

This must be done for the latitudes and also for the departures, 
and is entered in a column appropriated to each, called the ]N"orth 
and South correction, and the East and West correction ; the cor- 
rection, thus determined, must be placed, collaterally, with the 
distance to which it refers, without distinguishing as to JSTorth, 
South, East, or West. 

Having found the several corrections for each of the latitudes 
and departures, add them together severally, and see whether their 
total agrees with the whole error, and if so, proceed to allot the 
corrections. If the error be an excess of IN'orthings, substract 
each correction from its collateral Northing or add it to the colla- 
teral Southing : if an excess of Easting, add to the Westing and 



61 



substract from the Easting ; the corrected sums of the corrected 
latitudes and departures will then be found exactly to agree. 
We here subjoin an example: — 



.2 
m 


Bearings. 


Dis- 
tances. 


North. 


c 

o 
9 

o 


South, 


;-. 


East. 


o 
o 
;-! 

o 


West. 


^ 1 

o 
o 


E 








+ 








+ 






A 


63° 45' 


17-68 


7-83 


•040 


• . • 


, , 


15-85 


•080 


• • • 


. 


B 


67° 45' 


6-37 


2-61 


•014 


• • • 




5-6S 


•028 


. 




C 


47° 30' 


3-S6 


2-61 


•008 


. • • 




2-S4 


•016 


• • • 


, ^ 


D 


284° Ou' 


1-4-63 


3-54 


•033 


. . . 


. 


. . . 


, , 


14-19 


-066 


E 


212° 00' 


19-73 


. . . 




16-73 


.045 


. . . 


• • 


10-46 


•099 










+ 




— 




+ 






SuQiS, . . . 


62-27 


16-59 


•095 


16-73 
16-59 


•045 


24-37 


•124 


24-65 
24-37 


•165 




Difference, 


. . . 


• • • 




•14 








•28 





In the above, the error is -{- 'IJi in the South and + -28 in the 
West, we will now divide this error proportionately among the 
several distances, by the rule previously given, viz. : — As the sum 
of the distances : the whole error : : each distance : its parti- 
cular correction, or 

62-27 : -M : : 17-68 : "040 : : 637 : -014 : : 3-86 : "008 : : 14-63 : 
'033 : : 1973 : -045, for the ]S'orthings and Southings. 

And 62-27 : -28 : : 17-68 :' '080 : : 637 : '028 : : 3*86 : '016 
: : 14-63 : -066 : : lS-73 : -090 for the Eastings and Westings. 

It will be observed that the sum of the several corrections as 
above apportioned, amounts to '14 in the Xorthings and Southings 
or -095 + -045 = -14, and to -28 in the Eastings and Westings 
or -124 H- -1-56 = -28. This is the method of subdividing an 
error in theory, but in practice, an approximation is sufficient, the 
proportion of error to each line being made without reference to 
calculation, the error when it is below the maximum allowed, of 
one link in ten chains being equally divided between the two 



62 

columns of Northings and Southings and of Eastings and AVest- 
ings, and generally thrown into the longest lines. The example 
given, however, must not be understood as a specimen of the real 
extent of correction on such small distances, we have taken ample 
figures merely to serve the purpose of illustration. 

AV"e have omitted mentioning here the several methods given in 
other works on " Surveying by the Traverse System " of finding 
unknown distances, by addiug up the Northings and Southings, 
and the Eastings and Westings of a polygon, and applying the 
difference of the two severally, as the latitude and departure of 
the unknown line and thence finding the chain distance. Poly- 
gonometry as given in Hutton's Mathematics, Vol. 3, and other 
books, treat of these methods, and to them we refer the reader for 
further information. 



THE METHOD OP PLOTTING BY TRAVERSE. 

These difierences of latitude and departure, or distances on the 
meridian and perpendicular of each station from the preceding one 
are not only applicable to the proof of fieldwork, but are sub- 
servient also to the plotting and computation of the area of the 
Survey, which will now be explained. 

All the distances on the meridian of each station from the pre- 
ceding one, North or South, and all the departures of each station, 
from the preceding one, East or "West, can be referred to the 
meridian of the first station or starting point of the survey, viz., 
station A. Eor instance, on tlio meridian of A, ( Fig. Page 58, ) 
for the line AB, the distance North is A6 and the departure East 
is 6B ; on the meridian of B, for the line BC, the distance South 
is B/f, and the departure East is kC ; deduct the distance that B is 
North of A, from the distance that C is South of B, and we obtain 
the distance that C is South of A, or B^' — Ah ~ Ac; in like 



6S 

manner add the distance that is East of B, to the distance that B 
is East of A, and we obtain the distance that C is East of the 
meridian of A, or ^-^13 = cC 

Again, on the meridian of C for the line CD, the distance ISTorth 
is CI, and the departure East is ID, deduct the distance that C is 
South of A, from the distance that D is North of C, and we obtain 
the distance that D is JSTorth of A, or C^ — Ac = Ac?; in like 
manner add the distance that D is East of C, to the distance that 
C is East of A, and we obtain the distance that T> is East of the 
meridian of A, or ZD + cC =dJ). 

On the meridian of D, for the line DE, the distance North is 
Din and the departui'e TVest is wE, add the distance that E is 
North of D to the distance that D is North of A, and we obtain 
the distance that E is North of A, or Dm + Ac? = Ae : also, 
deduct the distance that E is "West of D from the distance that 
D is East of A, and we obtain the distance that E is East of A or 
dD — m'E =: eE, and so on all round the figure until arrived back 
at A, when the distance that A is North of J, the preceding 
station, deducted from the distance that J is South of A, or Aj — 
Js, and the distance that A is East of J, deducted from the distance 
that J is West of A, or J J — 5 A will leave no remainder, proving 
that the calculation has been correctly made. 

The line EG- it will be perceived crosses the meridian of A, in 
this case, it is only necessary to deduct the distance that E is East 
of A from the distance that Gr is "West of E, to obtain the distance 
thet a is "West of A or y^ - E/ = ^a. 

To plot, therefore, all these station points, draw a meridian line, 
and another perpendicular to it, representing the East and West 
direction. Eix on any point on this meridian line for the station 
A, lay off with a pair of common compasses and a scale of equal 
parts the distance Ab North of A, draw a lino parallell to the East 
and West line through the point h, lay off the distance 6B, East, 



64 

and join the points A and B, we thus obtain the bearing and 
distance of the line AB. 

Next lay off the distance Ac South of A, draw a line parallel to 
the East and West line, through the point c, lay off the distance 
cC East, and join the points B and C, thus obtaining the bearing 
and distance of the line BC. 

Then lay off the distance A.cl North of A, and with a parallel to 
the East and West line through the point d, lay off the distance dD 
East, join C and D, and we obtain the bearing and distance of the 
line CD, and so on all round the figure, observing that when the 
distances on the perpendicular are West of the meridian of A or 
starting point, they are laid oft" West on the plot. The reduction 
of the distances on the meridian and perpendicular of each station 
to the first station or starting point is therefore easily effected by 
a simple addition or substraction, and may be comprised in the fol- 
lowing rule. 

Rule. — When the distances run North of the first station, add 
them one to another, until they change to South, then deduct them 
one by one until the Southing exceeds the Northing, when deduct 
the latter from the former, changing the denomination to South ; 
all distances then going South, are added and those going North 
deducted, and so on, until arrived back at the original starting point. 

Likewise in the distances on the perpendicular, when the dis- 
tances run East of the meridian of the first station, add them 
one by one until they change to West, then deduct them until the 
Westing exceeds the Easting, when deduct the latter from the 
former changing the denomination to West ; all distances then 
going West are added, and those going East deducted, and so on, 
until arrived back at the original starting point. 

This method of plotting is by far the most perfect, and the least 
liable to error of any that has been contrived ; it may appear to 
require more labour, than the common method by angular protrac- 
tion or protraction by Bearings, on account of the computations 



65 

required, but these are made with so imicli ease and expedition by 
the help of Traverse Tables,* that this objection would vanish, even 
if they were of no other use than for plotting, but as we have 
already said, they are subservient also to finding the area, and 
which cannot be ascertained with equal accuracy in any other way ; 
when this is considered, it will be found to be attended with less 
labour on the whole than the common method. 

One great advantage in the above method of plotting is, that if a 
station be incorrectly plotted, it does not affect in the least, the 
correctness of the other stations, which is not the case when plot- 
ting with a common protractor by bearings or angles, where an 
error made in plotting one line is carried on through the series. 



THE TJK^IYEESAL THEOKEM. 

We must now look on the distances on the perpendicular above 
computed of each station, from the first in the series or starting 
point, as the sides of certain figures which multiplied into the dis- 
tances on the meridian between each station, will give certain 
products, from which the area of the figure is derived by an easy 
computation from the following : 

rXITEESAL THEOREM. 

If the simi of the distances of an East and West line of the two 
ends of each line of a Siorvey^from any meridian lying entirely out 
of, or running through the Survey, he multijolied hy the JSToethinq- 
or SozTHTSOr made on each respective line; the difference heticeen the 

* A. set of Traverse Tables has beai published by Major J. T. Boileau, Bengal 
Engineers, to every minute and degree of the quadrant, and these Tables are now in 
general use in the Revenue Surveys; we, therefore, refer the Surveyor to this work, 
in which he will find the method of using them fully explained and much valuable 
information regarding the application of the system to general purposes. A new 
edition of these Tables, carefully revised, is much wanted. 



66 



sum of the North Peoducts and the sum of the South Peoduots, 
ivill he double the area of the Survey. 

To explain this Theorem, it is necessary to take the three dif- 
ferent cases that present themselves separately, which are — 

1st. "When the meridian is to the West of the polygon and 
lying entirely out of the Survey, 

2nd. When the meridian is to the East of the polygon and 
lying entirely out of the Survey. 

3rd. When the meridian runs through the polygon a portion 
of the Survey lying East and West of it. 

Case 1st. 

When the meridian is to the West of the Polygon and lying entirely 

out of the Survey. 

Let ABCDEFA be any 

polygon, NS an indefinite 
straight, or meridian, line. 

Draw perpendiculars 0A, 
feB, cC, ^D, eE, /F, from 
the extremities of each side 
of the polygon, meeting 
^ the line JSTS, at a, b, c, d, e, 
f then will the distances ah, 
he, cd, de, ef fa, be the 
meridional distances corres- 
ponding to the sides AB, 
EC, CD, DE, EF, FA. 

If, therefore, we multiply 
the sura of the perpendi- 
culars at the extremities of 
each Northing or ascending 
side of the polygon by the 



i 




67 

meridional distance corresponding to that side, and place the 
products in one column, calling them North Products, and if we 
multiply the sums of the perpendiculars at the extremities of each 
Southing or descending side of the polygon, by the meridional dis- 
tance corresponding to that side, and place the products in another 
column, calling them South Products, then will the sum of the 
South Products deducted from the sum of the North Products be 
double the area of the polygon, that is : 



{aA + &B X rt& + cC + ^D X cc? + <?D + eE X de\ 
— /mT+Tc X he -{• eli +/F X e/ + /F + aA X /o, } 

will be double the area of ABCDEFA. 

Por, the sum of the North Products will be double the area of 
cCD^ec -\- aAW), and the sum of the South Products will be double 
the area of cCBAPEec + aA3h. 

But it is evident, from an inspection of the figure, that cCDEec 
- cCBAPE^c = ABCDEFA. If, therefore, the area of «AB6, 
which is common to the North and South Products be struck out, 
we have left cCD^ec - cCBAPEec = ABCDEFA, and conse- 
quently the difference of their doubles must be equal to double the 
area of ABCDEFA. 

Case 2ni). 

When the meridian is to the east of the polygon and lying entirely 

out of the Survey. 

Let AGrHIJA be any polygon, NS an indefinite straight, or 
meridian, line. 

Draw perpendiculars aK, gGr, hH, il, jJ, from the extre- 
mities of each side of the polygon meeting the line NS, at 
a, g, h, ^, j, then will the distances, ag, gh, hi, ij, ja, be the 
meridional distances corresponding to the sides AG, GH, HI, 
IJ, • JA. 

K 2 



68 




If we multiply ihQ sums of 
the perpendiculars at the extre- 
mities of each Northing or as- 
cending side, by the meridional 
distance corresponding to that 
side, and place the products in 
one column, calling them North 
Products, and if we do the 
same with each Southing or de- 
scending side, and place tlie 
products in another column, 
calling them South Products, 
then will the sum of the Nortli 
Products deducted from the 
sum of the ^outli Products be 
double the area of the poly- 
j gon, that is: 

-f /iH + a X u + il +iJ X ij\ 



— -[iJ + cA X ja -f rtA + ^G X a^ + gCi -i- /jH X Q^iX 

will be double the area of AG-HIJA. 

Por the sum of the North Products will be double the area of 
jViHGrAJ; and the sum of the South Products will be doable the 
area of j/iHI J/. 

But, it is evident from an inspection of the figure, that jhlllJj 
— jhUGAJj = AGtHIJA, and consequently the difference of 
their doubles must be double the area of AG-HIJA. 



Case 3ed. 

When the meridian runs through the Polygon. 

The annexed figure is a junction of the two last under a common 
meridian NS, passing through the point A, and consequently 
through the polygon. 



69 




J s 

To obtain the area of fhis polygon, in the same manner as in the 
last two cases, it is necessary to have a set of North and South 
products for that portion of the polygon lying to the East of the 
meridian line XS, another set for the portion lying West of the 
meridian, and again a third calculation for the line YGr, which, 
lying partly to the East and partly to the "West of the meridian, 
its two portions must be separately calculated. 

It will be observed that the only difference between the 1st and 
2nd cases is, that in the former, of the polygon lying to the £ast 
of the meridian, the South products are deducted from the North 
products, to obtain double the area, and in the latter, of the poly- 
gon lying to the Went of the meridian, the reverse takes place, viz., 
the North products are deducted from the South products to obtain 
double the area of the polygon ; this would be equally necessary 
in this polygon, but in practice, instead of having two sets of North 



70 

and South products, one set for the portion of a polygon, lying 
East of a meridian line, and another set for the portion lying "West, 
(in cases such as this where the meridian line runs tlirough the 
polygon) it is usual to reverse the products of that portion of the 
polygon to the West of the meridian line, and enter them in the 
same columns as the products of the portion of the polygon to the 
East of the meridian line, i. e., that all products of sides running 
Norths to the West of the meridian, are placed in the column of 
Soutli products, and vice versa, all products of sides running South, 
to the West of the meridian are placed in the column of North 
products. 

Two columns are thus sufficient, for if in the 1st case, the South 
products deducted from the North products give double the area 
to the East of the meridian, and in the 2nd case, the North pro- 
ducts deducted from the South products give double the area to 
the West of the meridian, and the North products and South pro- 
ducts of the 2nd case, are changed and applied as South products 
and North products in the 1st case, or vice versa, we shall obtain 
the same result, as if we had two sets of North and South pro- 
ducts. 

It only remains, therefore, to explain how the area is obtained of 
that portion of the polygon lying East and West of the meridian 
or on the line EGr. 

Produce Efto x and Gy to y and draw Qcx andyE parallel to NS. 

It is evident, from an inspection of the figure, that if the area 
of the rectangle ^fgy to the East of the meridian be found, and 
placed in the column of South products, it being a Southing or 
descending side, and the area of the rectangle y^&y, be found and 
placed in its column of South products, it being also a Southing 
or descending side to the West of the meridian, (supposing us to 
have two sets of North and South products,) and we reverse the 
latter and place it in the column of North products in the Traverse 
Table, the line EG- will have a North and a South product too, one 



71 

product being the area West of the meridian, the other the area 
^ast of the meridian, but to facilitate calculation and simplify the 
work, it is better to deduct the lesser product from the greater, 
and carry the difference to the column in which the excess is. 
The same result, however, is obtained by taking the difference 
between the perpendiculars East and West of the two points, and 
multiplying it by the meridional distance corresponding to the 
line, placing the product in its proper column North or South, as 
the case may be ; in the present instance it is 

gy X Fy = Area of YfgTj or South Product to Ihe East of the Meridian 
and gG x Gx = „ foGg ,„ „ „ West „ 

or reversed North ,, ,, East ,, 

The difference of the two would be carried to the South products, 
the excess being South, we should obtain the same result however, 
if we take the difference betfveen i/g and ^Gr and multiply by Fy, 
whick is the usual method in the Traverse Table. 

Having now explained why these products are called JSTorth and 
South products, and also shown that the difference between them 
gives double the area of the polygon, we will exemplify how they 
are obtained by a reference to the diagram, page 69. 

"We have already said, that to obtain the area we must look on 
the distances on the perpendicular as computed from the first sta- 
tion or starting point, as the sides of certain figures which multi- 
plied into the^ distances on the meridian, will give the North and 
South products above alluded to, for instance : — • 

On the line AB, the distance on the perpendicular 5B, multi- 
plied into the distance on the meridian Ah, will give double the 
area of the triangle ABb, a North product, the line AB running 
Northward. 

On the line BC, the sum of the distances on the perpendicular at 
each end, or 5B + cC multiplied into the distance on the meridian 
he, wdll give double the area of the Trapezoid hcCB, a South pro- 
duct, the line BC, running Southward. 

On the line CD, the sum of the distances on the perpendicular 



72 

at each end or cC + dD multiplied into the distance on the meri- 
dian cdj will give double the area of the Trapezoid dcCD, a North 
product, the line CD runniDg Northward, and so on, until we 
arrive at the last product on the line J A. 

The North and South Products being then respectively added 
up, the difference between the two sums will be double the area of 
the Survey, the half of which will give the area in acres and deci- 
mal parts of an acre. 

"We may here observe that in all the works in which the Univer- 
sal Theorem has been treated of, for the ascertainment of areas, the 
meridian has been assumed as lying entirely out of the Survey, but 
this is contrary to practice. The meridian of a village circuit must 
naturally pass through the first station of a Survey, at the point 
where the instrument is first set up, and except in very peculiar 
figures, this causes a portion of the figure to fall on hoth sides of 
the meridian. 

To assume a meridian to pass entirely out of the Survey, it is 
necessary to go out of our way, and from its extreme Easting or 
Westing to adopt a quantity greater than either, and so calculate 
the length of each co-ordinate from this assumed distance, at the 
same time that it involves the necessity of an extra calculation for 
finding the area. There is no possible advantage in this method, 
as the same result is obtained by making use of the co-ordinates 
East and "West of a meridian running through the Survey, giving 
considerably less figures in the calculation and consequently less 
labour in deducing the products. 

To follow therefore the simplest and most natural course as met 
with in daily practice, must be the most advantageous, and it is not 
only so as regards the area, but likewise in respect to the plotting ; 
to take the meridian passing through the first station, the pro- 
traction is at once easily and simpl}'- laid down, without the 
necessity of fufcher calculation and more inconvenient lengths of 
scale and compass. 



73 



An assumed meridian out of the Survey ia &till more at variance 
with systematic precision and progess, whereas in extended opera^ 
tions many villages are plotted on the same sheet of paper, and 
where each circuit must be built on its own meridian passing 
through the first station. On the Indian Eevenue Surveys, there- 
fore, the shorter and more practical method is pursued, and when 
it is considered that from 1 500 to 2000 circuits are on an average 
annuallv surveyed by each party employed, a faint idea may be 
gleaned of the labour saved by the improvement above speci- 
fied. 



THE TEATEESE TABLE. 

Hayikg in the last few pages given an explanation of the 
Traverse system of surveying from its commencement, viz. : — the 
angular work and measurements in the field, to the finding the 
area of the land surveyed, it only remains to simplify the process 
which has taken some pages for explanation, by embodying the 
whole in a table for the purpose, and calculating the polygon given 
in page 69. In the adjoining table 

Col. No. 1 contains the letters representing the stations of the Survey. 



5 
6,7 
8,9 

a, a, a, a, 

10 

11 
12 



the angles as observed in the Field. 

the corrections made in those angles, to prove them by 

Eule, page 51. 
the bearings of the several lines deduced, as per Rule, 

page 54. 
the distances as measured in the Field. 
the distance on the meridian between every two stations, 
the departure of each station from the meridian of the 

preceding one. 
the corrections applied to each calculation, to prove the 

Survey correct, as per Rule, page 60. 
the distances on the meridian of each station from the 

tirst in the series, 
the departure of each station from the first in the series, 
the sums of each pair of co-ordinates obtained from 

Column 12, to be multiplied into Columns 6 and 7, 

when the respective products are placed in Columns 13 

and 14. 



7i 



•S3J0V ui spupo.tj tunos ^ 


CO CO " CO 

• (M • . CO W • • 

* TTi * ' t^ r-( (M * * CO 

I— 1 <— 1 



p 

CO 


"saiov "! sp-npojj "q^io^si 


k5 


(M CO CO Ci 

■^ • "? 'P • • • <r' ■? • 

CO '00 * ' ■ i 00 * 

-* C^ CO CM 


CO 

■?* 



-ipjo-oo SB aBinoipuadiacI 
aqj no saouB^sip 9i[i jo suing 




CMCMOCMCit-OCOCOO 
'^ (M- >p -Tf.-^ ^ -^ t- 

oo'—i'^'ffleoco"*— luo 

'-^ CO CM --< <-! "M (M 


- 


•sai-tas 3i[} ui uon 
-■Bjs }s.Tg sqj mo.ij uopins 
qoBa je sjni.iBcIap .10 acina 
-ipuacljad aqi uo S30Lre:)siG[ 





C-lOOOCOOOCOiOO 

^v vs -^ i^- XI cp '^ tx: t^ 

to itM OC CiO t^ •<* CI '.'5 kO 


: 


JO UBTpUaiU 91^ uo S30UT3^SI(J 


5 


CO-^OCit— OOCOOKJO 
CO 7* t^ "*■ -p <^0 00 C3 <M 

irjCNij-j'^acooic^coioO 

— C^ r-t ,-( CM 

5^< W Z C/2 


• 


re 

_o 
■^ 

p< 

>-< 


« 


m 


•uoipgjio^ 


I 


* I I • • « • • 


1 p 




03 




CO Cft ■"# » 

* * ' cj <M 4f «b ' * 


00 

CO 
CO 


•uoTpgijtoo 


X 


p • tp p 


+ 9 


00 


M 

CO 




<M 00 ^ CO 10 


C30 

CO 

CO 


'3 

V-l 
F=i 

C 

to 
■u 
u 

C 

s 


•uoipajjoo 


X 


. • I * I . I . 


+ 9 





i2 


7-47 

4-42 
4-69 

14-83 
13-28 


9 


•1totp9.iJ.03 


1 


• • p 


1 2 




5 


C3 tH C5 no 00 
CO ^C>0 1~-. * ■•<*i * ^t^) 



9 


on; ut pgjnsBaoi se saouB^STQ 


to 

H 
u 


iO<=>COO(M>iOCOO<M — 

COi— CiCO»OOiO-^-'^t» 


: 


OOCiUOCO-^COOsCtDt^ 


•S9{Sav 
moij p9onp9p sSuunag 





(MCOCO'«*l«Oi^O0C30.— 
r-iCMCO-— -lOO'-'OC—l 

00.— 1 10 --OOCO CM ■»*»--. 

LO -^ CM — ' CI ^rf CO to -^ -^ 

— ' CO — CM CO (M •— 1 


CM 





•U0l}09IIOJ 


X 


II ! I I I " I . CM 


1 

■ppi^r 

9in ut U9:^[Bl St' S9[§uv 





— 10 — CM-— 1 — ocoooco 
,-<,_,,^T^o(MTpO>-im 

j^ 1— ' »o CO ri< <^^ a: oc 

(N »-> CM CM — " — ' 









1 -SUOIlE^g 

1 




<220QK!faOEi-(^<: 






10 w 

o as 






X 



bo 



O C>3 



+ 



E< 



The following simplification of the . method of calculating 
columns 4, 6, 7, 8, 9, 10, 11, 12, 13, and 14 will, it is hoped, place 
the matter beyond doubt ; and by comparing each given quantity 
with the diagrams, the mode of obtaining its corresponding result 
will be easily understood. 



To obtcnn Column 4 of tlie Table from Column 2. 



G-iven the 
Astronomical 

Then, 

Bearg. AB 50° 
„ BC 140 
„ CD 21 
,, DE 315 
„ EF 191 
,, FG 248 
„ GH 333 
„ HI 202 
IJ 144 
„ JA 47 



Bearing of the line AB, 50® 12' X. E. as found by 
observation, or otherwise. (Diagram, page 55.) 



12' + 
23 + 
33 -f 

14 + 

56 + 

57 + 
13 + 
5S 4- 
01 + 
19 + 



ZB 270°11'=32C^ 23' — 1S0°=140' 



,,G 61 ;0=201 

„ D 113 41=135 

5, E 56 42 =371 

„ F 237 01=423 

,,G264 21=513 

„ H 49 40-382 

„ I J21 03=324 
„ J 83 1S2 = 27 

„ A182 53=230 



33 -^180 =251 
14 4-180 =315 

56 —ISO =191 

57 —ISO =248 

18 —180 =333 

58 —ISO =202 
01 —180 =144 

19 —180 = 47 
12 —180 = 50 



23'S.E. 


or 


Bearg 


. BC 


33 N.E. 


or 


jj 


CD 


14 N.W 


. or 


j> 


DE 


5 6 S.W. 


or 


)> 


EF 


57 S.W. 


or 


5> 


FG 


18 N.W 


or 


)> 


GH 


58 S.W. 


or 


••> 


HI 


01 S.E. 


or 


>> 


\i 


19 N\E. 


or 


J> 


J A 


12 N.E. 


or 


J> 


AB 



To oltain Columns 6, 7, 8. and 9. 

Biile. — As Eadius : Distance : : Cosine of Bearing : Latitude 
and : : Sine of Bearing : Longitude or Departure. 

Latitude =^ Distance X Cos. Bearing ; or 
Departure = Distance x Sin. Bearing. 

See Diagram, page 58. 
Line AB Lat. A 6 = AB Cos. A 



, BC 


)> 


B 


k 


- BC 


»> 


B 


, CD 


j> 


C 


I 


= CD 


)) 


C 


, DE 


j> 


D 


m 


= DE 


M 


D 


, EF 


»' 


E 


n 


= EF 


J> 


E 


, FG 


9} 


F 


y 


= FG 


Jt 


F 


, GH 


•y 


G 


V 


= GH 


)> 


G 


, HI 


J5 


H 


Q 


= HI 


V 


H 


, IJ 


»> 


I 


r 


= IJ 


'> 


I 


, JA 


J» 


J 


s 


= JA 


}> 


J 



Long 


h 


E = AB £ 


ine 


A 


M 


h 


C = BC 


M 


B 


>J 


I 


D= CD 


>J 


C 


)i 


m 


E = DE 


J> 


D 


J> 


n 


F = EF 


>> 


E 


J) 


y 


G - FG 


!» 


F 


5 ) 


V 


H r: Gil 


>> 


G 


'J 


a 


I = HI 


:» 


H 


J5 


r 


J = IJ 


j> 


I 


J» 


s 


A =JA 


!> 


J 



76 



JExamples of Columns 6, 7, 8 and 9. 
On the Line AB given, Bearing N.E. 50^ 12' Distance 8.35. 

Cosine. Sine. 

Bearing 50°12' 9-806254 9-885521 

Distance 8-35 0-921686 0'9216S6 



•727940 5-34 Lat. '807207 = 6-4 1 D3p. 

The above example bj Major Boileau's Tables (See Note, page 
65). 

Latitude. Departure. 

Bearing 50°] 2' 5-J2 .... 800 ... . 6*15 

•18 ... . 30 ... . -23 
Distance 8-35 -04 ... . 5 .... -03 

5-3< 6-41 

To ohtain Column V^from Columns 6 and 7. 

Line AB Dist. on Merd. of A to B or A6 = N 5-33 
„ BC „ „ B „ C „ m = S 7-47 

S 2-14 Diflf. /Distance on Meridian of 
„ CD „ „ C „ D „ C« = N 14-84 \ A to C or Ac 



N 12-70 „ r „ 

„ DE „ „ D „ E „ Dm = N 979 \ of A to D 



or Ad 



N 22-49 Sum r 

\ of. 



„ EF „ „ E „ F „ En = S 4-42 \ of A to E or Ae 

N 18-07 Difif.r „ 

„ FQ „ „ F „ G „ Fj/ = S 4-G9 \ of A to F or A/ 

N 13-38 „ / „ 

„ GH „ „ G„H„G;) = N 945 \ of A to G or A^ 

N 22-83 Sum/ „ 

„ HI „ „ H „ I „ Hg = S 14-83 „ \ of A to H or AA 

N 800Diff./ „ 

,, IJ „ „ I « J „ Ir = S 13-28 „ \ of A to lor A» 

S 5-28 „ / „ M _ ,t 

., JA „ „ J M A „ J« = N 5-28 ,, \ of A to J or hj 

0-00 



77 



To obtain Column 11 from Columns 8 and 9. 
Line AB Departure of A to B or 6D = E 6-42 



It BC o 

„ CD „ 

» FG 

>i GH „ 

>» HI „ 



B ,, C ., iC = E 6.11 



E 12-60 Sum = Dep. of A to C or eO 
C „ D „ /D = E 5-86 



E 18-46 „ „ „ A „ D „ dD 

D „ E „ ^E = W 9-70 



E 8-76 Diff. „ „ A „ E „ Ee 
E „ F „ nF = W 0-93 



E 7 83 „ „ „ A „ F ,,/F 

F „ G „ </G = W 12-19 



W 4-36 „ „ „ A „ G „ 

G „ H „ pH = W 4-74 



W 9-10 Sum „ „ A „ H „ AH 
H „ I „ ql = W 6-28 



W 15-38 Sum = Dep. of A to I or tl 
I „ J „ rj = E 9-63 



W 5-75 Diff. „ „ A „ J „ jJ 
J „ A „ sA = E 5-75 



0-00 

To oltain Column 12 from Column 11. 

Line AB Departure of A to B or &B - E 6-42 



„ cd{ 



,. DE 



EF 



A „ B „ iB = E 6-42 
A „ C „ cC = E 12-60 



E 19-02 Sum of Co-ordinates. 



A „ C „ cC = E 12-60 
A „ D„(iD = E 18-46 



E 31*06 



A „ D „ c?D = E 18-46 
A „ E „ eE =: E 8*76 



E 27-22 „ 



A „ E „ eE = E 8-76 
A „ F „ /F = E 7-83 



E lC-59 „ 



„ fg| 



„ GH 



„ Hl{ 



„ IJ 



, JA 



78 



A ., F „ /G := E 7-83 
A „ G„^G = W 4-36 



E 3-47 Diff. of Co-ordinates. 



A „ G „ ^G =: W 4-36 
A „ H„ hU =z W 9-10 



W 13-46 Sum 



A „ H „ ^H = W 9-10 
A „ I „ a = W 15-38 



W 24-48 



A „ I „ il - W 15-38 
A „ J „ jj Z W 5-75 



= W 21-13 



A „ J „j3 = W 5-75 



To ohtain Columns 13 and 14i,from the mvMi^lication of Column 12, 

into Columns 6 and 7. 





















North Prod. 


South Prod. 




AB 




6B 


X ^>A 


or N 


6-42 


X 


E 


N. E. & S. W. 


N. W. & S. E. 


ine 


5-33 ~ 3-42 


)) 




BC 


&B 


-}- cC 


X ho 


„ S 


19-02 


X 


E 


7-47 = „ 


14-20 




CD 


cC 


+ rfD 


X erf 


„ N 


31-06 


X 


E 


14-84 - 46-08 


j> 




DE 


d\) 


+ eE 


X rfe 


„ N 


27-22 


X 


E 


9-79 = 26-63 


)) 




EF 


eE 


+ /F 


X ef 


„ s 


16-59 


X 


E 


4-42 = „ 


7-33 




FG* 


y'F 


- ^G 


X /5- 


V S 


3-47 


X 


E 


4-69 = „ 


1-63 




GH 


.-G 


+ /iH 


X gh 


„ N 


13-46 


X 


W 


9-45 = „ 


12-71 




HI 


/iH 


+ il 


X /« 


„ s 


24-48 


X 


VV 


14-83 = 36-29 


j» 




IJ 


il 


+ iJ 


X ^/ 


,, s 


21-13 


X 


w 


13-28 - 28-06 


>) 




JA 




iJ 


X iA 


M N 


5-75 


X 


w 

Sl 


5-28 = „ 


3-03 




ms, . , 140-48 


38-90 




















38-90 





DifF. . . 101-58 Acres = double 
the area of ABCDEFGHIJ A. 



* The Co-ordinates here changing from East to West, or crossing tire Meridian of 
Station A, the dijference between the pairs instead of the sum is taken, by which 
means the area of the Parallelogram F.r G^, is properlj' balanced, the portion lying 
to the West of the Meridian of A, being cancelled by a portion on the East side, 
leaving a difference in favour of the latter, and which accordingly remaining East, and 
multiplied by a Southing, forms a South Product. 



79 

Consequently, 



(bB X b^ -{- cC ■]- dD X. cd -\- dD ■\- eE X de + h\i + il X hi + il + J J Xij) — 
(bE+TU X be + eE +/F X ef + fF-gG X fg + gG + hti X gh+jJ xJA) 
= Double the area of the figure ABCDKFGHIJA, Page 69. 

We have now endeavoured to explain the Traverse Table, and 
though all the operations which have been given at length for the 
sake of explanation appear laborious, they are performed with 
the greatest facility, the whole, with the exception of Colamns 6, 
7, 8 and 9, being obtained by a simple addition or Subtraction, 
and Columns 13 and 14 by multiplication. Such is the traverse 
system of surveying, and to use Mr. Adams'* words, "the superior 
accuracy and ease with which every part of the process is performed, 
cannot, it is imagined, fail to recommend it to every practitioner." 



Geometrical Essays, by George Adams, 1313. 



80 



CHAPTER V. 



THE POCKET SEXTANT 



The Pocket Sextant combines numerous valuable properties : 
it measures an angle to one minute of a degree, requires no sup- 
port but tbe hand, may be used on horseback, maintains its adjust- 
ment long, and is easily re-adjusted when put out of order. It will 
determine the latitude by a meridian altitude to one minute ; and 
an approximation may even be made with it to the longitude, by 
means of lunar observations. Parther, it is very portable, forming 
when shut up, a circular box under three inches in diameter, and 
only li inches deep. 
o 




The above figure represents the instrument screwed to its box, 
for convenience of holding in the hand, and with the telescope 



81 

drawn out. A, is the index arm, haying a vernier adjusted to the 
graduated arc B, which latter is numbered to 140°, but the sextant 
will not measure an angle greater than about 125*^. The index is 
moved by the milled-head C, acting upon a rack and pinion in the 
interior. Two mirrors are placed inside ; the large one, or index 
mirror, is fixed to, and moves with the index : the other, called the 
horizon glass, is only half- silvered. The proper adjustment of the 
instrument depends on these glasses being parallel when the index 
is at zero— while they are, at the same time, perpendicular to what 
is termed the plane of the instrument, represented by its upper 
surface or face. To observe whether the instrument is in perfect 
adjustment, remove the telescope by pulling it out, and supply its 
place with a slide far the purpose, in which is a small hole to look 
through : then place the index accurately at zero, and direct the 
instrument, holding it horizontally, towards the sharp angle of a 
building, not less than half a mile distant, applying the eye so as to 
see both through the hole in the slide, and also through the un- 
silvered part of the horizon glass : the same object ought then to be 
so reflected from the index mirror fcothe silvered part of the horizon 
glass, as to seem but one with the object seen direct : if such be 
not the case, a correction becomes necessary, which is thus per- 
formed : D is a key, removeable at pleasure, that fits two keyholes, 
the one at «, the other at h. Apply this key at a, and gently turn, 
until the reflected object, and the one seen direct, seem but as one. 
The glasses are then parallel. 

The next point is to examine whether the horizon glass is per- 
pendicular to the plane of the instrument. For this purpose, hold 
the sextant horizontally, and look at the distant horizon ; then, if 
any adjustment be wanted, two horizons will appear, or the reflec- 
ted one will be higher or lower than the one seen direct ; should 
this be the case, apply the key at &, so as to bring the two horizons 
together. It must be observed that the large, or index mirror, be- 
ing correct by construction, it can want no alteration. 

M 



82 



By looking at the sun, we can always satisfy ourselves with 
respect to the adjustments; the telescope has a dark glass at the 
eye end, and with this on, we have only to place the index at zero, 
and using the telescope, to look at the sun — when, provided the 
instrument is in exact adj ustment, one perfect orb only will be 
seen. If the reflected image project beyond the other, then correc- 
tion is necessary. The full moon will answer as well as the sun for 
this purpose ;'but the dark glass at the eye end of the telescope must 
then be removed. The instrument is provided with two other dark 
glasses, which sink out of the way by raising two little levers at/*. 

It has been mentioned above, that for trying the adjustments of 
the sextant, an object must be half a mile off; this is on account of 
what is called the parallax of the instrument, occasioned by the ne- 
cessity of placing the eye of the observer on one side of the index 
mirror. Could we look from the middle of it, there would be no 
parallax ; which is the angle subtended by the point of vision, and 
centre of the index glass, when observing any near object ; conse- 
quently, as the distance of an object is increased, this angle dimi- 
nishes, and at length, becomes as nothing when compared with it. 
Half a mile is considered sufficient for all error to vanish, but at 
half that distance, it is scarcely perceptible. 

To take an angle, the observer looks either through the telescope, 
or hole in the slide (having previously raised the levers of the dark 
glasses at /), at the left hand object, holding the sextant horizon- 
tally in his left hand ; with his right, he turns the milled head C, 
until the other object, reflected from the index glass, appears upon 
the silvered part of the horizon glass, exactly covering or agreeuig 
with the left hand object, seen direct through the unsilvered por- 
tion of the horizon glass ; the angle is then obtained by the ver- 
nier to one minute. 

If the required angle be a vertical one, the sextant is held in a 
vertical position, by the right hand, while the left turns the milled 
head C, until the object is brought down to the horizon. 



83 



When the altitude of a celestial body is taken at sea, it is 
brought down, as the term is, to the natural horizon, and the 
measure of the angle, or height of the object, is read off upon the 
graduated are ; but on land, the natural horizon can seldom be 
used, on account of its irregularity ; recourse is then had to, 
what is called, an artificial horizon, such as a vessel containing 
water, mercury or other fluid. The observer then places himself in 
a situation, to see the reflected image of the sun, or other body, in 
the fluid : he has only then to bring down the image, as reflected 
from the index glass, until it reaches its reflection in the fluid : tlie 
altitude will then be ^aZfthe number of degrees, indicated by the 
graduated arc, subject to certain corrections, not necessary to be 
explained here. 

The reason of only taking half the number of degrees^ [will^be 
seen from the following explanation :— ' 

Let AB represent the 
surface of the quicksil- 
ver contained in a wood- 
en trough, whose plane is 
continued to C; DEF, 
the roof, in which are 
fixed two plates of glass, 
DE and EF, whose sur- 
faces are plane and paral- 
lel to each other, and Q 
the sun at S, whose alti- 
tude is required. Now 
the ray SH, proceeding 

from the sun's lower limb to the surface of the quicksilver, will be 
reflected thence to the eye in the direction of HGr, and the upper 
limb of the sun's image, reflected from the quicksilver, will appear 
in the line GH, continued to E-; and it is a well known principle 
in catoptrics, that the angle of incidence, SHA or SHC, is equal to 

M 2 




84 



*s 



the angle of reflectioUj GHB ; and as the angle AHE, or CHE, is 
the opposite angle of GHB, it is, therefore, equal to it, and to the 
angle SHC, the altitude of the sun's lower limb above the horizon- 
tal plane : so that, if we suppose the angle SHE, measured by a 
sextant, to be SO"", the altitude of the sun's lower limb will be 40°, 
subject to the corrections, as above. 

The principle of the construction of the sextant may Be ga- 
thered from the following demonstration. 

Let ABC represent a Sextant, having 
an index, AG, (to which is attached a 
mirror at A,) moveable about A as a 
centre, and denoting the angle it has 
moved through, on the arc, BC : also let 
the half-silvered (or horizon) glass, a h, 
be fixed parallel to AC ; now a ray of 
light, SA, from a celestial object, S, im- 
pinging against the mirror, A, is re- 
flected off at an equal angle, and striking 

the half-silvered glass at D, is again reflected to E, where the eye 
likewise receives, through the transparent part of that glass, a direct 
ray from the horizon. Then the altitude, SAH, is equal to double 
the angle, CAG, Jiieasured upon the limb, BC, of the instrument. 

Por the reflected angle, BAG, (or DAP) = the incident angle, 
SAT, and the reflected angle iDE == the incident «DA — DAE ~ 
DEA, because al i^ parallel to AC. jN'oav, HAI = DFA r= 
(FAE + EEA), and DAE, being equal to DEA, it follows that 
HAI = (DAE + EAE). From HAI and (DAE -f FAE) take 
the equal angles, SAI and DAF, and there reiuains SAH = 
2 FAE, or 2 GAC ; or, in other words, the angle of elevation, 
SAH, is equal to double the angle of inclination of the two mirrors, 
DGA, being equal to GAC. 

Hence the arc on the limb, BC, although only the sixth part of 




85 



a circle, is divided as if it were 120°, on accouut of its double 
being required as the measure of CxlB, and it is generally extend- 
ed to 140° 

The chief and indeed onl}^ objection to the sextant as a survey- 
ing- instrument, arises from the ang-les taken with it not being: 
always like those measured by the theodolite and compass, liori- 
zontal ones. If the theodolite be set truly level, we can take 
angles all round its circle, no matter whether one object be high 
and another low, and these angles will be what are termed hori- 
zontal or azimuthal angles ; so that Avere we to take angles from 
object to object and complete the circle, the sum of all these angles 
ought to be 360^, or the measure of a circle. But if the angles 
were to be measured by a reflecting instrument, they would not 
amount precisely to 360°, unless taken upon a perfectly level 
plane ; owing to this circumstance, that to take an angle by the 
sextant, the two objects have to be brought into contact, viz,. — the 
reflected one and that seen direct, it is necessary for the observer 
to hold his instrument, not strictly horizontal, but in the plane of 
the two objects, or in such a position as will enable him to form 
the contact ; and, therefore, if one point is elevated very much 
above the other, the sextant must be held at a corresponding in- 
clination Avith the horizon. Angles so taken require a reduction 
to horizontal ones. But as the sextant is never used to lay down 
points for a iTigonometrcal Survey of importance, it rarely occurs 
that the reduction is required ; indeed, to eff'ect it with accuracy 
is attended with considerable difiiculty, as the angles of elevation 
or depression must be known, a matter always difiicult of attain- 
ment with the sextant. It is better to avoid, if possible, the 
necessity of this reduction by selecting stations neither much 
elevated or depressed. 

The height and distance of objects, as walls or buildings, whether 
accessible or otherwise, may be obtained in a very simple and ex- 



86 



peditious manner with the sextant, by means of the little table 
below : — 



Multiplier. 


Angle. 


Angle. 


Divisor 


1 . . . 


. . . 45° 00' . . . 


. . . . 45° 00' . . . 


... 1 


2 . . . 


... 63 26 ... . 


... 26 34 . , . 


... 2 


3 . . . 


... 71 34 . . . 


... 18 26 . . . 


... 3 


4 . . . 


... 75 58 . . . 


... 14 02 . . . 


... 4 


5 . . . 


. . . 78 41 . . . 


... 11 19 . . . 


... 5 


6 . . . 


. . . 80 32 . . . 


... 9 28 . , . 


... 6 


8 . . . 


. . . 82 52 ... . 


... 7 08 . . . 


... 8 


10 . . . 


. . . 8i 17 ... . 


... 5 43 . . . 


... 10 



■ 



Make a mark upon the object, if accessible, equal to the height 
of your eye from the ground. Set the index to one of the angles 
in the table, and retire on level ground, until the top is brought by 
the glasses to coincide with the mark ; then, if the angle be greater 
than 45°, multiply the distance by the corresponding figure to the 
angle in the table ; if it be less, divide — and the product, or quoti- 
ent, will be the height of the object above the mark. Thus, let 
EB be a wall, whose height we want to know ; and 26° 34' the 
angle selected. Make a mark at D 
equal to the height of the eye ; then 
step back from the wall, until the 
top at E is brought down by the 
glasses to coincide with the mark : 
measure the distance AB, namely, 
from your station to the wall, and 

divide that distance by 2, the figure corresponding to 26"* 84', this 
will give the height DE, to which BD must be added. 

The par alia jc of the instrument exerts an influence on measure- 
ments of this kind, from the object being near. To correct it, we 
have only to ascertain its amount, by placing the index at zero, 
and looking through the instrument at the top of the wall ; when, 
if influenced by parallax, it will appear as a broken line ; but by 
moving the index a little way on tbe arc of excess or to the left of 



c,-"' 



87 



zero, the broken line will reunite, and the adjustment be effected. 
When any quantity is taken thus on the arc of excess, the amount 
must be deducted, when setting the instrument to any of the 
tabular angles. 

"When the object is inaccessible — set the index to the greatest of 
the divisor angles in the table, that the least distance from the 
object will admit of, and advance or recede, till the top of it be 
brought down by the sextant to a level with the eye ; at this place, 
set up a staff, equal to the height of the eye. Then set the index 
to one of the lesser angles, and retire in a line from the object, till 
the top be brought to coincide with the staff, set up to indicate the 
height of the eye ; place a mark here, and measure the distance 
between the two marks ; this, divided by the difference of the 
figure opposite the angles used, wdll give the height of the object 
above the height of the eye or mark. For the distance multiply 
the height of the object by the numbers against either of the angles 
made use of, and the product will be the distance of the object 
from the place where such angles was used. 

The above will be understood better by means of a diagram. 
Let AB be a wall, not to be approached nearer than C ; and that 
we find, upon trial, that this distance admits of our using the angle 




45° ; assume a point E on the wall, as the height of the eye ; then 
the index being set to 45°, fix yourself so that the glasses shall 
bring the top A to coincide with E. At this point, place a staff, 



88 



CGr, equal to the height of the e3'e. Now select any one of the 
lesser angles from the tables — 18° 2G', for instance, and retire 
until the point A agrees with the top of the staff CGr, which occurs 
at P. Place a mark at F, and measure the distance from P to Gr ; 
which, divided by 2, the difference of the numbers opposite to the 
angles used, will give AE— to which add BE = CGr, the height of 
the eye, and the total height AB is obtained. Then," for the dis- 
tance — the height AE, multiplied by 3, its corresponding figure, 
will give the length EE ; and AE multiplied by 1, will, in like 
manner, give GrB = AE in this instance. 

Horizontal distances, as well as heiglits, may be found by means 
of this table where the ground is level ; but as we may not always 
be able to measure in a direction at right angles to the distance we 
wish to ascertain, (as in the acompanying diagram), I prefer using 
a more independent method. Suppose AB to be the breadth of a 
rirer which it is 
desired to find ; 
send a flag in 
any direction a- 
long the bank 
of the river to 
C, and with the 
sextant measure 
the angle ABC, 
(A being any 
object on the 
opposite bank 

of the river), set the index to half the supplement of the angle, ABC, 
and proceed in the direction BC, until the glasses of the instrument 
shew a flag staff at B, and the object A, in contact. Suppose D to 
be this point, then will BD be equal to AB, the breadth sought. 
The reasoning of the above is very simple, the supplement of ABC 
is EBA, which is equal to BDA + BAD, and since we make 




89 

BDA half EBA, tlierefore, BAD is also equal half EBA, and, 
therefore, BDA = BAD, and therefore BD = BA. 

These methods of determining heights and distances are valuable, 
as the operations are speedily performed, and with tolerable accu- 
racy ; while it enables us to dispense with logarithmic tables and 
trigonometry. 

The pocket sextant is very useful, when taking offsets ; set the 
index to 90°, and walk along the station line ; then, when you wish 
to ascertain at what point any mark or object becomes perpendicu- 
lar to the station line, you have only to look through the sextant 
at the left hand object, and move forward or backward until the 
two objects, namely, the offset mark, and that on your station line, 
are brought to coincide. Or, if you wish to lay off a line at right- 
angles to another, send your assistant with a staff in the required 
direction, and having set the index at 90°, cause him to move right 
or left until his staff and your other mark are made to agree. 

An instrument which has been made specially for the above 
purpose and which, where attainable, has quite superseded the use of 
the cross staff, is the " Optical Square." It is made of brass, and 
contains the two principal glasses of the Sextant, viz., the index 
and horizon glasses, fixed at an angle of 45° ; hence while viewing an 
object by direct vision, any other, forming a right angle with it at 
the place of the observer, will be referred by reflection, so as to 
coincide with the object viewed. This instrument has the advan- 
tage of great portability, not being larger than an ordinary sized 
hunting watch. 



N 



90 



CHAPTER VI. 



LEYELLII^G. 



Leyelling- is the art of tracing a Hue at the surface of the earth, 
which shall cut the directions of gravity everywhere at right 
angles. If the earth w^ere an extended plane, all lines representing 
the direction of gravity at eveiy point on its surface would be 
parallel to each other ; but, in consequence of its figure beiug that 
of a sphere or globe, they everywhere converge to a point within the 
sphere which is equi- distant from all parts of its surface ; or, in 
other words, the direction of gravity invariably tends towards the 
centre of the earth, and may be considered as represented by a 
plumb-line when hanging freely, and suspended beyond the sphere 
of attraction of the surrounding objects. 

Eor the better elucidation of the subject, I consider it best to 
give the reader some idea of the method of levelling before enter- 
ing into minute detail of the instruments employed, and of their 
adjustments, which latter will be better understood, when some no- 
tion is formed of the ends which they are intended to accomplish. 
Therefore, in the following explanations in speaking of a spirit level, 
the reader, w^ho is unacquainted with this description of instru- 
ment, must imagine a telescope similar to that of the theodolite, and 
mounted in somewhat the same way, the horizontal wire of which, 
when the instrument is levelled, will trace out a right line parallel 
to the horizon. 



91 




In tlie above- diagram let the straigJit line AB represent the sur- 
face of the earth, upon the supposition of its being an extended 
plane, the direction of gravity at the points, A, I, and B, would be 
represented] bj the lines, AC, ID, and BE, all parallel to each 
other, and at right angles to the horizontal line AB. JS'ow, if the 
surface was undulatorj, as shown by the curved line AB, and it 
was required to make a section representing it, an instrument 
capable of tracing out a line parallel to the horizontal line AB, (as 
a spirit-level,) might be set up any where on the surface, as at I, 
and staves being placed or held along the line, as at a, h, c, d, Sfc.^ 
the different heights above the ground where such staves wxre in- 
tersected by the lines so traced out, would at once show the relative 
level of all these points, Avith regard to the horizontal line, as a 
datum or standard of comparison. 

But as the earth is a globe, its circumference must be circular, 
as IKL, in the annexed figure : the straight line AB, will, there- 
fore, not represent the surface of the earth, but the sensible horizon 
of an observer stationed at the point I, to which point it is 
a tangent, being at right angles to the radius of the circle 
(or semi-diameter of the earth) IC. A line which is parallel 
to the sensible horizon of the observer, is the line traced 
out bv our spirit-levels, and is a tangent to the earth's surface 
at that point only where the instrument is set up : thus, A B 

is 2 



92 



N B 




13 a tangent at I, and DE a tangent at P; such being the fact, 

the difference of level 
between any two 
points cannot be de- 
termined by simple re- 
ference to a horizon- 
tal line, since every 
point on the surface 
of the globe (however 
near to each other) 
has a distinct horizon 
of its own. 
If the earth were everywhere surrounded by a fluid at rest, or 
that its surface was smooth, regular and uniform, every point 
thereon would be equally distant from the centre ; but, in conse- 
quence of the undulating form of the surface, places and objects 
are differently situated, some further from, and others nearer to 
the centre of the earth, and consequently, at different levels. The 
operation of levelling may, therefore, be defined as the art of finding 
how much higher or lower any one point is than another, or, more 
properly, the difference of their distances from the centre of the 
earth. 

Eeferring to our last figure, we have seen that the line AB is a 
true horizontal or level line at the point I, but being produced in 
the direction A or B it rises above the earth's surface; and, al- 
though it may appear to be level, as seen from I, yet, it is above 
the true level, (which is represented by the circumference of the 
circle,) at every other point, and continues to diverge from it, the 
further it is produced ; at Gr, the apparent line of level, as the 
horizontal line AB is called, is above the true level, by the distance 
Gil, and at M by the distance MN, the difference leing equal to 
tlie excess of the secant of the arc of distance alove the radius of the 
earth. 



93 



The 



difference, GrH or MN, between the true and apparent 

level may be thus found. Put t 
in the following diagram for 
the tangent IH, o' for the ra- 
dius, cl of the earth, and ^ 
for GH, the excess of the se- 
cant of the arc of distance 
above the radius ; IH being 
considered as equal to IG- ; 
then 




■f2 



(r + ccy = r- + t 

or 5^' + 2rx + x"" = r' + f 

and2r.r + x' = f 

or (2r + ^^) X = i' 

But, the diameter of the earth 2r is so great with respect to 

the quantity (x) sought at all distances to which a common 

levelling operation usually extends, that 2r may be taken for 

2r-\-x without sensible error ; we then have 

2rx ^ f 



and 



2/' 



Or in tuords : — The difference (x) between the true and apparent 
level is eqiialtothe sguare of the distance (f) divided hy the diameter* 
of the earth {2r) ; and, consequently, is always iirajportional to the 
square of the distance. 

The mean diameter of the earth is 791G miles and the excess of 
the apparent above the true level for one mile .-. = >,. ^^^ of a mile, 
or 8"004i inches, at two miles it is four times that quantity or 32-016 
inches, and so on increasing in proportion to the square of the dis- 
tance. 

If we reject the decimal •OOJi and assume the difference between 
the true and apparent level for one mile, to be exactly 8 inches, or 
two-thirds of a foot, there arises the following convenient form for 



9-i 



computing the correction of level due to the curvature of the earth, 
for distances given in miles, viz, — — , D being the distance in miles. 
A very easily remembered formula, derived from the above for 
the correction for curvature in feet is two-thirds of the square of 
the distance in miles ; and another, for the same in inches is the 
square of the distance in chains divided hy 800. 

Prom the above it will be seen that the effect of taking the 
apparent instead of the true level, is to make the level of the point 
observed fo, lower with reference to the point observed from, 
than it really is, so that the correction for curvature must be added. 
Eor instance, suppose I find the apparent level of a point G50 feet 
distance to be + 6*52 feet ; that is to sa}^, 6'52 feet higher than the 
point from which I observed, the correction for curvature for that 
distance is '01 ; and, therefore, the true level of the point is + 6'53 ; 
but supposing the apparent level be — 652 ; then the true level 
will be — 651. 

But this effect, due to the earth's curvature, is modified by 
another cause arising from optical deception. This second correc- 
tion, viz. — terrestial refraction, has the contrary Q^ect oi elevating 
the apparent place of any oljject above its real place. The rays of 
light bent from their rectilnear direction in passing from a rare 
into a denser medium, or the reverse, are said to be refracted, and 
this causes an object to be seen in the direction of the tangent to 
the last curve. Every difference of level, accompanied as it must 
be with a difference of density in the strata of the atmosphere, will 
have, corresponding to it, a certain amount of refraction, and as the 
curve described by each ray of light is concave next the earth ; the 
tangent to the curve will lie above it, and consequently, the object 
will appear more elevated than it really is. 

A simple rule to correct the error occasioned by refraction, is to 
diminish the effects of the earth's curvature by one-seventh of 
itself, this is not quite correct, but will be found sufiTiciently so in 
most cases. 



95 

A table of curvature and refraction and both combined, in which 
form they are generally given, will be found at the end of the 
chapter. It must be borne in mind that if these corrections are 
applied directly to the reading of the staff, they must be used in an 
opposite way to what they are when applied to the difference of 
level, viz., the curvature must be subtracted, and the refraction 
added ; for the lower the point observed to, the greater is the read- 
ing on the staff; if, therefore, we were to add the correction for 
curvature to the reading on the staff', we should be making that 
point lower still, instead of counteracting the effect which has 
already made it too low. 

But in ordinary levelling, i. e. with a spirit level and two staves, 
these corrections are very seldom applied in practice, for under a 
distance of 700 feet, they are inappreciable, and the injurious effects 
that they might have in a succession of stations during a long 
day's levelling, are counteracted by placing the instrument mid- 
way between the staves, the effect of curvature is thus altogether 
removed as well as that of refraction, as the latter will affect both 
observatious alike, unless under peculiar circumstances of weather, 
over which tlie observer has no control. The fact of placing the 
instrument equally distant from each staff, has also other advan- 
tages which will be pointed out hereafter. 

The method of proceeding in levelling is shewn in the annexed 
figure . 

, — ~ZlA ^ hr~ 

A C ^ ^ & 

Suppose it were required to find the difference of level between 
the points A and Gr ; a Staff is erected at A, the instrument is set 
up at B, another staff at C, at the same distance from B, that B, is 
from A, and the readings of the two staves are noted. The instru- 
ment iy then conveyed toD, and the staff which stood at A, is now 



90 



removed to E, the stafi C, retaining its former position, and from 
being the forward staff at the last observation, is now the back 
staff : the readings of the two staves are again noted, and the in- 
strument removed to F, and the staff C, to the point Gr, the staff at 
E, retaining the same position now becomes in its turn the back 
staff, and so on to the end of the work, which may thus be extended 
many miles : the difference of any two of the readings will show 
the dift'erence of level between the places of the back and forward 
staff; and the difference between the sum of the back-sights, and 
the sum of the forward-sights will give the difference of level be- 
tween the extreme points, thus : 





E 


Jack-Sights. 




Fore-Sights 






Ft. Dec. 




Ft. Dec. 


A, and C, 




10-46 




11-20 


Cj 5? E, 




11-33 




8-00 


E, „ Gr, 


Sums 


7-42 




7-91 




2921 


2711 






27-11 







Difference of level, 2*10 
Bhowing that the point Gr, is 210 feet, higher than the point A. 
The foregoing process is called compound levelling. The following 
is an example of simple levelling, being preformed at one operation 
and therefore subject to the correction for curvature and refraction 
to obtain a correct result. 




97 

If it were required to drain a Jheel A, by making a cut to a 
stream at B, a distance of 20 chains : let a level be set up at C, and 
directed to a staff held upright at the edge of the water at B. The 
horizontal line CD represents the line of sight which would cut the 
staff at D, the reading being 17'4;4 feet ; the height of the instru- 
ment above the ground was- 4i feet, and the depth of the Jheel 10 
feet ; therefore the difference of level between the bottom of the 
Jheel and the surface of the stream was as follows : 

Ft. Dec. 

Reading of the staff 1 7*4i 

Height of instrument i 4'00 

Depth of Jheel lO'OO 

Curvature and Refraction for 20 Chains. 1 
see Table, at end of chapter, ,,.,,./ * ' 

• 14-03 



Difference of level, ». 3-4] 

LETELLI^■& STAVES. 

Levelling staves are of various patterns, but thev may be divided 
into two classes. Levelling staves with, vanes ; and those on which 
the scale of feet is painted in a bold conspicuous manner, which ena- 
bles the observer to note the reading himself. In the former, the 
vane, which is a piece of wood painted half black and half white, 
and which slides on the staff, is moved up or down by an assistant 
according to the signals of the observer, until the horizoDtal wire of 
the telescope bisects it, when the height should be read off by the 
staff-man. A second vane is placed at a height of six feet from the 
first ; when the readings are greater than six feet, the upper vane is 
used and six feet is added to the reading on the staff. 

In levelling with the theodolite and for contouring, this description 
of staff may be used with advantage, but for ordinary levelling, it is 
greatly inferior to the staves which can be read by the observer, 
especially in this country where the staff-man cannot be trusted to 
give the reading ; the staff has to be brought for the leveller to read, 
thus much time is lost, and the leveller is always in the hands of 
the staff-man, who by accident or on purpose may shift the vane ; 

o 



9S 

also it takes some time to adjust the vane to tlie proper height, 
especially if the staff-man is not accustomed to the work. The only 
advantage of these staves is, that the vane can be bisected at a greater 
distance than the figures of the others can be read. 

The second description of levelling staves are made, either in one 
piece, or two or more pieces, which have joints like those of a fi.sh- 
ing rod, and fit one into the other (a very objectionable pattern), 
or telescopic. I prefer the first for use, about twelve feet long. The 
telescopic staves are more portable, and equally good if they do not 
get out of order, but they are very liable to do so in this country 
from the swelling of the wood ; they are, moveover, three times the 
expense ofthe simple staff in one length. Several different ways of 
marking levelling staves have been tried, to enable them to be clearly 
read. The best is that pattern designed by Mr. Conybeare, the 
Engineer of the Vehar Water-works, which can be read at a greater 
distance than any other staff I know of. 

Having got, so far, I will now proceed to treat of levelling instru- 
ments and their adjustments. 

theYievel. 

The accompanying figure represents the Y level. 

A, is an achromatic telescope, resting upon two supporters, which 
in shape resemble the letter Y, and are consequently called the ys. 
The lower ends of these supporters are let perpendicularly into a 
strong bar, which carries a compass box, C. This compass box is 
convenient for taking bearings, and has a contrivance for throwing 
the needle off its centre, when not in use. One of the Y suppor- 
ters is fitted into a socket, and can be raised or lowered by the 
screw B. 

Beneath the compass box, which is generally in one piece with 
the bar, is a conical axis passing through the upper of two parallel 
l^lates, and terminating in a ball supported in a socket. Immedi- 
ately above this upper parallel plate is a collar, which can be made 



99 



to embrace the conical axis tightly, by turning the clamping screw 
E, and a slow horizontal motion may then be given to the instru- 
ment by means of the tangent screw D. The two parallel plates, 




are connected together by the ball and socket already mentioned 
and are set firm by four mill-headed screws, which turn in sockets 
fixed to the lower plate, while their heads press against the under 
side of the upper plate, and thus serve the purpose of setting th© 
instrument up truly level. 

Beneath the lower parallel plate is a female screw, adapted to 
the stafi'-head, which is connected by brass joints with three ma- 
hogany legs, so constructed, as to shut together, and form one round 
staff, a very convenient form for portability, and, when opened out, 
to make a firm stand, be the ground ever so uneven. 

The spirit level, I I, is fixed to the telescope by a joint at one end, 
and a capstan-headed screw at the other, to raise or depress it for 
adjustment. 

The telescopes of levelling instruments are provided with cross 
wires, fitted to a diaphragm, as before described for the theodolite 
and moveable by the same means. The cross wires in the diaphragm 
of the level are usually arranged as shewn in the diagram ; the 
horizontal wire marks the intersection of the horizontal visual ray 

o 2 



100 




with the staff; the two vertical wires serve to direct the telescope 

so that the staff shall be seen 

between them, and thus be in 

the axis of the lenses; and by 

their means the observer can 

judge whether the staff is being 

held vertical. 

The Y level has. three adjust- 
ments, viz.: — 

1st. — T7ie line of Collimation ; 
i. €. to bring the horizontal wire 
into the line of collimation of the 

telescope. First, let us see what would be the effect on levelling 
operations if this were not done. 

Let A B, represent a section of the plane of the wires, and C 
the proper position of the horizontal wire ; the telescope 
is set truly horizontal, and on its being turned in the 
direction of a staff, a portion of the latter comes into the 
field of view, forming a picture on a smaller scale than 
the original, at A B ; then the division of the staff cor- 
responding to the point C is on a level with the axis 
of the telescope ; but suppose the horizontal wire is at D, it 
will cut two staves, placed at equal distances from the instrument, 
at points differing from the true level points, by as many divisions 
of the staff as are included between C and D, and since the staves 
are at equal distances, the scale of the picture formed in the teles- 
cope, and, therefore, the number of divisions between C and D, will 
be the same in both cases ; and the reading on each staff will differ 
from the true level point by the same quantity, they will therefore 
give the true differences of level between the two places. But if 
the staves are at unequal distances, the reverse of this takes place, 
for the scale of the picture, and, therefore, the number of divisions 
included between C and D alter as the distance of the staff is 






101 

altered ; if, therefore, the horizontal wire be not at the point C, the 
readings obtained where the staves are at unequal distances from 
the instrument, will not give the true difference of level between 
the two places. 

To effect this adjustment, make the horizontal wire coincide with 
some well defined distant object, then invert the telescope by 
turning it half round in its Ys, and if the wire has moved off the 
object, bring it back, half by the adjusting screws above and below 
the telescope, and half by the foot screws ; repeat this until the wire 
coincides with the object in both positions of the telescope, when 
the adjustment will be complete. 

2nd. — To adjust the lubhle tube. Open the clips of the Ys and 
placing the telescope parallel to two of the foot screws, by their 
means bring the bubble to the centre of its run ; now reverse the 
telescope in the Ys by turning it end for end, and if the bubble 
still remain in the centre, it is in adjustment ; but if not, bring it 
back to the centre, half by the capstan-headed screw at one end of 
the bubble tube, and half by the foot screws. Eeverse the teles- 
cope again, and if still not correct repeat the operation until it is 
so. The bubble tube will now be parallel to the line of coUimation. 

8rd. — To set the telescope at right angles to the horizontal axis ; 
so that when this axis is vertical, the line of coUimation may de- 
scribe a horizontal plane. 

To do this, bring the telescope over two of the foot screws and 
by their means bring the bubble attached to the telescope to the 
centre of its run, turn the telescope half round on its axis, and if 
the bubble should then leave the centre, bring it back, half by the 
adjusting screw B which raises or depresses the Y, and half by the 
foot screws ; repeat this till perfect, and then turn the telescope a 
quarter round, and by means of the third foot screw bring the bub- 
ble again to the centre of its run. The bubble should now remain 
in the centre during a complete revolution, and the instrument will 
be in adjustment. 



102 



The Y level is more easily adjusted, but from their superior 
compactness, increased optical power, and greater stability of the 
adjustments, Grravatt's and Troughton's levels are more generally 
used. 

geavatt's oe the dtjmpt leyel. 

This instrument is furnished with an object glass of large aper- 
ture and short focal length, and sufficient light being thus obtained 
to admit of a higher magnifying power in the eye-piece, the advan- 
tages of a much larger telescope are obtained, without the incon- 
venience of its length. 




The diaphragm is carried by the internal tube a a, which is 
nearly equal in length to the external tube. The external tube 
T T, is sprung at its aperture, and gives a steady and even motion 
to the internal tube, which is thrust out, and drawn in. to adjust the 
focus for objects at different distances by means of the mill-headed 
screw A. The spirit level 1 1, is placed above the telescope, and 
attached to it by capstan-headed screws, one at either end, for the 
purpose of adjusting it. 



103 

A cross level ^, is placed upon the telescope at riglit angles to 
the principal level I I, by v^hich we are enabled to set the instru- 
ment up at once, with the axis nearly vertical. It is only to be 
used for thus levelling the instrument, approximately. The mirror 
M mounted upon a hinge joint, can be placed at the end of the 
level, 1 1, so that the observer, while reading the staff, can at the 
same time see that the instrument retains its proper position, a 
precaution by no means unnecessary in windy weather, or on bad 
springy ground. The telescope is attached to the horizontal bar 
by capstan-headed screws, B B, space being left between the bar 
and the telescope for the compass box C. D is the clamping screw, 
and E the tangent or slow motion screw; but sometimes these 
levels are made without either, as shewn in the Troughton level. 

Almost all levels, of every description, are now made with three 
foot-screws instead of the old parallel plate screws. 

As the telescope of this is fixed and not moveable in Ys, it is not 
possible to collimate it the same way as the Y level. I have al- 
ready explained that if the horizontal wire be not in the line of 
collimation, then even though the axis of the telescope be horizon- 
tal, the wire will not trace out a liorizontal line ; but more than this 
the wire will not even trace out a right line ; for example, (see fig. 
next page,) with the level at A, if the wire be fixed on d of the staff 
B, then it will not cut the point E, on the staff C, directly in a line 
with d, but will cut some point above or below as /"or g. The rea- 
son of this is as follows, let A repre- 
a | — — . — : — \c sent the proper position (i. e. in the 

line of collimation) of the horizontal 
wire, and CA the direction of the axis of a pencil of light passing 
though the object glass and coming to its focus at A. Then the 
axis of the tube of the telescope being set truly horizontal, the 
line AC, is also truly horizontal, and every point bisected by 
the horizontal wire will be situated on the prolongation of the 
line AC. 



lOi 



Suppose now the position of tlie diaphragm carrying the wires to 
have become deranged, so that the horizontal wire is moved to B, 
then every point intersected by this wire will be on the prolonga- 
tion of the line BC. Buit in order to see clearly things at different, 
moderate distances through a telescope, the object glass must be 
di'awn in or thrust out, the point C will thus alter in position, and 
therefore the prolongation of BC will not be the same line for short 
as for long distances. But if the horizontal wire be at A, the point 
C moving along AC, or the axis of the telescope^ the prolonga- 
tion of AC will always remain the same. 

On the above has been founded G-ravatt's method of collimating. 
To discover if the wire traces out a right line or no. 

Pirst find the horizontal line, a' h" c", then if the point e be in a 




straight line with the points d and a', the triangles a' l" d and 
a' c" e, will be similar, and e c" will be to d V as a c" to a' h" . To 
find this horizontal line, having chosen a tolerably level piece of 
ground, place the staves A, B, C, at equal distances, about 150 or 
200 feet apart, on pegs driven in the ground ; now place the level 
at D, half way between A and B, and having levelled it, read the 
staff A, then turn it round, and if necessary, having re-levelled it, 
read staff B, then since the level is equally distant from A and B, 
the points a and h tbus obtained will be on the same level, what- 
ever may be the errors of adjustment in the instrument* : proceed 
similarly at E, half way between B and C, and we shall obtain the 

• The reason of this is that the enof in adjustment will aflfect the readings on each 
stafF equally , to find the difFerence of level we subtract these readings, one from the 
other, the error will therefore be eliminated. 



105 



two points y and c on the same level. The difference of the two 
readings on staff B is 5 })\ add this (with its proper sign) to the 
readings on staff 0, and this gives the point g\ "We now have the 
three known points, a h c', equidistant from the earth's centre, and 
a line passing through them will be horizontal. 

Now take the level to A, and placing it so that the height of the 
axis of the telescope may "be measured by the staff A when per- 
pendicular, read the staffs B and C. From A.a\ the height of the 
axis of the telescope, subtract A« the height of the point a, this 
will give aa, apply this with its proper sign to B6 and Co', and 
we obtain the two points h" , c" on a level wdth a\ !N'ow subtract 
W)" and Qc" from the last readings on B and C, respectively, and we 
get h"d and c"e, the former of which should be half the latter, since 
ah" = J a'c" ; but, if not, the wire is out of adjustment and re- 
quires correcting. To do this, loosen the upper and tighten the 
lower screw holding the diaphragm plate, or vice versa^, and take a 
another reading to B and C, (and if the instrument has been moved 
at all, A.a' must be re-measured ; this will affect B5" and Co" which 
must be corrected accordingly,) again subtract from these read- 
ings B5" and Cc", and if Vd is not now half c" e, the operation 
must be repeated until it is so. 

The second adjustment is to put the level tube parallel to the line 
of collimation. After perfecting the above adjustment, if the instru- 
ment be not moved, all we have to do to effect this, is to direct the 
horizontal wire to the point c'^ (by means of the foot screws) on 
the staff C, and since this point being on the same level as a', the 
line of collimation is horizontal, bring the bubble to the centre of 
its run (by the adjusting screws at one end of the tube) and it will 
be horizontal, and, therefore, parallel to the line of collimation. 



* Care must be taken to make a note of wliat was done in each trial, such as " gave 
top screw one quarter turn to the right or left," for guidance in perfecting the ad- 
justment, after having seen whether the previous operation has made it better or 
worse. 



106 



The third adjustment, viz., setting the line of colliraation per- 
pendicular to the axis, is performed in the same way as in the Y 
level. Instead of there being a screw at one end though for the 
purpose of making this adjustment, there are at each end three 
screws, secured bj a covering plate which must first be removed ; 
the two outside ones pusb up, and tbe middle one pulls down, that 
end of the telescope. 



T E OTTGHTOITS LEVEL, 

In tbis level, tbe telescope T, rests close down upon the horizontal 




bar 55, the spirit level II, is permanently fixed to tbe top of the 
telescope, and does not, therefore, admit of adjustment, and the 
compass box C is supported over the level by four small pillars at- 
tached to the horizontal bar. This construction makes the in- 
strument very firm and compact. This instrument may be adjusted 
in the same manner as that described for the Dumpy level, witb the 
exception that it does not allow of the second adjustment, the level 
tube as before described, being permanently fixed by the maker. 
When the axis of collimation has been made horizontal, then if the 
bubble is far from the centre, the instrument must be rejected as 
a bad one ; but if it be only a little out, the error can be calculated 



107 



by means of the scale on the bubble tube when the staves are placed 
at unequal distances, in which case only, the readings would be 
aiFected by it. Having discovered by Grravatt's method that the 
instrument is a good one, i. e. that the level tube has been correctly 
fixed, should the instrument again get out of adjustment, we can 
rectify it by the following method, which when practicable, is both 
simple and accurate. Drive t^fo stakes into a pond or other con- 
venient piece of water, at about a distance of one hundred yards 
one from the other, with their heads level with the surface of the 
water, the stakes should be so placed that you can put your instru- 
ment in a line with them and close to one of them. Now, having 
levelled your instrument, and staves being held upon the stakes 
read the nearest one, then if the further one does not give the same 
reading, the horizontal wire must be moved up and down, until 
it does so, by means of the diaphragm screws. This method does 
not apply to the Dumpy level, as we have no means of ascertaining 
if the level is correct ; and, therefore, of levelling it. 

Having now given a description of the ordinary levelling instru- 
ments, I will return to the practice of levelling. 

The following will explain the method to be pursued in levelling 
a tract of country. 

In the first instance the stafi'-holder must place his stafi* on the 
Bench mark* from whence the levels are to commence. The Sur- 
veyor must then set up his spirit-level in the most suitable spot 
which presents itself, from whence he can have an uninterrupted 
view, not only of the stafi" at; the back station, but also for a consid- 
erable distance in the direction he wishes to carry his levels. The 

* In the practice of levelling, it is usual to leave at convenient intervals, what 
are called Bench marks; these mostly consist of permanent objects, such as stumps 
of trees, rrilestones, &c., on which it is usual to cut a distinguishing mark, that it 
may be known hereafter. Their use is chiefly for future reterence, in the event of 
its being necessary, either to check the levels by repetition, to change the direction 
of the line of levels from any point, or to take up and continue the levels at the 
commencement of a day's work, a Bench mark having been left at the close of the 
day preceding. 



108 



station selected should not in any cas9 exceed 4 or 5 chains, for 
when long distances are taken, unless both the back and forward 
stations are equally distant from the instrument, errors w411 gradu- 
ally creep in, which in a long series of levels, are liable, by their 
accumulation, to be of serious consequence. 

The proper station being determined on, and the level adjusted 
for observation*, it must be directed to the back staff and the foot 
and decimal fraction of a foot, with which the central part of the 
horizontal wire appears to be coincident, noted w^ith all possible 
exactness, and entered in the proper column of the Pield or Obser- 
vation Book ; as soon as it is registered, look to see that the spirit 
bubble has not returned from its central position, and then repeat 
the observation, to ensure that no mistake has been made in noting 

it. 

The back observation being made, turn the Telescope round in 
the forward direction ; then look at the spirit bubble, and if it has 
at all changed its position, by receding towards either end of the 
tube, bring it back to the centre by the parallel plate screws, then 
observe what division on the staff is intersected by the cross-wire, 
and enter the reading in the proper column of the Pield-book. 
Having entered it, verify it by a second observation, which will 
complete the first levels. The first levels being completed, the 
Surveyor, passing the man who holds the forward staff, proceeds to 
some convenient spot to set the instrument a second time, which, 
as before remarked, should not be more than 4 or 5 chains distant ; 

* The Level must be adjusted for observation in the following order; First draw 
out the eyepiece of tlie Telescope till the cross-wires are perfectly defined ; then, di- 
recting it to the staff, turn the focussing screw on the side of the Telescope, till 
the smallest graduations on the statF are likewise clearly distinguishable; that 
these two adjustments be very carefully and completely performed, is of more con- 
sequence than is generally supposed, for upon them depends the existence or non- 
existence of parallax, to remove which has already been explained at page 42. The 
ebove operations having been gone through, bring the spirit bubble into the centre 
of its glass tube, which position it must retain unmoved in every direction of the 
instrument ; this is accomplished in the same manner as in the Theodolite, by 
bringing the Telescope successively over each pair of the parallel plate screws, 
and giving them motion, screwing up one, while uniicrevving the other to an equal 
extent. 



109 



the man who held the staff at the back station lilvewiso proceeds 
still further onwards to take up a new station, and as "nearly as 
possible at the same distance from the instrument, as the instru- 
m.ent is from the staif, which has now become the back station. 
The instrument is tben again adjusted, and the same process follow- 
ed as above described, until arrived at the end of the series. 

The foregoing description of the method of taking levels is 
general, and applies equally to every kind of levelling operation. 

The following is the form of Field-book used for entering the 
observations, &c. 



.2 

CO 

;-< 


Back. 


Forward. 


■ (5 




0) 

> 

s 

p5 


Reniarks. 


6 

o 

a 

13 


1 


■CD 

c 

P5 


.s 




C 
P 






o / I 






o / 












1 


200 


271-30 


5-85 


3-50 


87-00 


200 


2-35 




100-00 




2 


200 


269-30 


4-75 


4-50 


91-00 


200 


•25 




102-35 


4-3 
S-l 


o 


200 


267-15 


2-88 


8-75 


89-50 


262 




5-87 


102-60 


o 


4 


100 


231-30 


0-18 


9-63 


48-00 


100 




9-45 


96-73 




5 


150 


221-30 


0-06 


6-80 


16-30 


200 




6-74 


87-28 




G 


200 


181-30 


4-71 


2-18 


301-30 


200 


2-53 




80-54 




7 


200 


96-30 


11-50 


0-18 


262-30 


200 


11-32 




83-07 


d 


8 


300 


74-00 


11-00 


1-48 


248-45 


300 


9-52 




94-39 


5 


9 


4U 


71-30 


7-32 


1123 


184-00 


413 




3-91 


103-91 


Q 



Sometimes a column is inserted between the back and forward 
readings, in which is noted the height of the instrument at each 
station, thus furnishing the level of the ground at that point also. 

It is usual to refer all levels to some fixed datum line, which is 
easily recognisable, suck as the mean level of the sea ; in England, 
Trinity Higli AVater Mark ; in Calcutta, the sill of the stone on the 
Tide Guage at Kyd's Dockyard, &c. The reduced levels shew the 
height above or below such datum line. Where this is not done, 



110 



ifc is usual for convenience sake to take an imaginary datum line, 
some even number of feet above or below the first, or any other 
station, and reduce all the levels to this, the object is to save the 
necessity of using signs (+ or — ) in the reduced levels. 

"Where levels are made for the formation of a section it is neces- 
sary that the distance between the levelling staves be measured, as 
well as the bearing observed of each staff to enable the Surveyor to 
plot and draw the section, but in running or check levels, there is 
no necessity for the chain or compass, the object of check levels 
being only to obtain the difference of level between certain inter- 
mediate and the extreme points of the section previously made, to 
check its accuracy. It is also immaterial by what route we pro- 
ceed from one point to another, so that such spots may be selected 
for the stations as are most convenient for the purpose, and may 
afford opportunity of checking any intermediate points on the sec- 
tion line. The Pield-book required therefore for check levels, is 
merely a simple entry of back and fore-sights, the difterence of the 
sums of which will be the difference of level between the extreme 
points of the Section line. 

In plotting sections of levels a larger scale is generally used for 
the vertical than for the horizontal distances. Ey this means 
space is economised in the length of the section, and the slopes of 
the ground, especially in roads, railroads, canals, &c., the depth of 
cuttings, height of embankments, &c., are shewn with much greater 
clearness than if the two scales were equal. Ten to one is a con- 
venient proportion between the scales. 

LEVELLIlS^a "WITH THE THEODOLITE. 

In taking sections across broken irregular ground intersected by 
ravines, this system of operation is recommended as being much 
more easy and rapid than tracing a series of short horizontal lines 
with the spirit level. Where, however, this latter instrument can 
be used with tolerable facility, it should always be preferred. Level- 



Ill 

ling for sections by angles of elevation and depression -with the 
theodolite is thus performed. 

The instrument is set up at one extremity of the line previously 
marked out by banderoles or long pickets at every change of the 
general inclination of the ground ; and a levelling-staif, with the 
vane set to the exact height of the optical axis of the Telescope, 
being sent to the first of these marks, its angle of depression or 
elevation is taken, and by way of insuring accuracy, the instrument 
and staff are then made to change places, and the vertical arc being 
clamped to the onean of tlie two readings^ the crosswires are again 
made to bisect the vane. The distances may either be chained before 
the angles are observed, marks being left at every irregu-larity on 
the surface where the levelling staff is required to be placed ; or 
both operations m.ay be performed at the same time, the vane 
on the staff being raised or lowered till it is bisected by the wires 
of the Telescope, and the height on the staff noted at each place. 

The accompanying sketch explains this method : — A and B are 
the places of the instrument, and of the first station on the line, 
where a mark equal to the height of the instrument is set up ; 
between these points the intermediate positions », 6, c, f?, for putting 
up the levelling staff are determined by the irregularities of the 
ground. The angle of depression to B is observed, and if great 
accuracy is required, the mean of this and the reciprocal angle of 




elevation from B to A is taken, and the vertical arc being clamped 
to this angle, the Telescope is again made to bisect the vane at B. 
On arriving at B, after reading the height of the vane at a^ J), c, Sfc, 



112 



and measuring the distances A a, ^c, the instrument must be 
brought forward, and the angle of elevation taken to C, the same 
process being repeated to obtain the outline of the ground between 
B and C. In laying the section down upon paper, a horizontal 
line being drawn, the angle of elevation and depression can be 
protracted, and the distances laid down on these lines ; the respec- 
tive height of the vane on each staff being then laid off from these 
points in a vertical direction, will give the points a, h, c, Sfc, marking 
the outline of the ground. A more correct way of course is to 
calculate the difference of level between the stations, w^hich is the 
sme of the angle of depression or elevation to the hypothenusal 
distance AB considered as radius, allowing in long distances for 
curvature and refraction. 

Instead of only taking the single angle of depression to the 
distant Station B, and noting the heights of the vane at the inter- 
mediate Stations, a, h, c, Sfc, angles may be taken to mark the same 
height as the instrument set up at eacli of these intermediate points, 
which will equally afford data for laying down the Section ; but 
the former method is certainly preferable. 

The details may be kept in the form of a Field-book, but for this 
species of levelling the measured distances and vertical heights can 
be written without confasion on a diagram, leaving the corrections 
for refraction and curvature (when necessary) to be applied when 
the section is plotted. 

"Where a number of cross sections are required, the Theodolite is 
particularly useful, as so many can be taken without moving the 
instrument. It is also well adapted for trial sections, where minute 
accuracy is not looked for, but where economj^, both of time and 
money, is an object. 

The theodolite is likewise used in running check levels, to test 
the accuracy of those taken in detail with a S2nrit-level. Eecipro- 
cal angles of elevation and depression, taken between bench marks, 
whose distances from each other are known, afford a proof of the 



113 

general accuracy of the work ; and if these points of reference are 
proved to be correct, it may safely be inferred that the intermedi- 
ate work is so likewise. 

CO^'TOURIKG. 

The last description of levelling by the spirit level to be noticed, 
is the method of tracing in'strumentally, horizontal sections termed 
"contours," either round a group of isolated features of ground for 
the formation of plans for drainage, sanitary, railway, or other en- 
gineering purposes ; or over a whole tract of country with a view of 
giving a mathematical representation of the surface of the ground 

in connection with a national, or other extensive and accurate sur- 
vey. 

Contour lines give a most prefect delineation of the ground, and 
they are the only part of a survey which will remain unaltered in 
the lapse of ages, hills and valleys being much more permanent 
things than houses, roads and boundaries, which cease to give accu- 
rate information in a few years and require revision at a great cost. 

It would be useless expense to increase the number of contour 
lines on mountain ground where no probable demand either for 
roads or drains exists ; and on the other hand in districts which 
are nearly level, contours only at great diiference of altitude would 
be of little practical utility. 

In waste lands, contours tend to a knowledge of the best mode 
of improvement, as the levels are connected with each other 
throughout the country, and referred to the sea as a datum line. 
As a general system, however, contouring can scarcely be said to 
be applicable to India, where the mountains are inaccessible and 
for the most part untrodden, and the wastes impenetrable and im- 
pervious, from the denseness of the jungle and rankness of the 
vegetation. The undulations and round smooth downs of Englaud 
are here wanting, and the vast extent of the country leaving but 
few points fixed by the great triangulation, the operation, so simple 



114 

ou the Ordnance Survey of England, would be one of much diffi- 
culty in this country, where there is so little to mark the inequali- 
ties of the surface until the stupendous hills I'ise suddenly and pre- 
cipitously above the general level. 

A few remarks on the system, however, which has become so 
common in England, w^ill not be misplaced. 

" The method of tracing these contours in the field is thus per- 
formed. Banderoles or long pickets are first driven, one at the 
top and another at the bottom of such slopes as best define the 
ground, particularly the ridge lines and watercourses ; should no 
such sensible lines exist, they must be placed at about equal 
intervals apart, regulated by the degree of minutise required, 
and the variety in the undulations of the surface of the ground. 
A short picket being driven on the level of the intended upper 
(or lower) line of contours, and in line between two of the ban- 
deroles, the level is placed so as to command the best general view 
of this first line and adjusted, care being taken that its axis is not 
so low as to cut the ground below the picket (or so high as to be 
above the top of the levelling-staff", if the lower contour is the 
first traced) ; the staff is then placed at this picket, and the vane 
raised or lowered till it is intersected by the cross-wires of the 
Telescope, the staff is then shifted to another point on about the 
same level, and in the line between the next two pickets, and the 
staff itself moved up or down the slope till the vane again coin- 
cides with the cross-wires, at which spot another picket is driven. 
This operation is continued, till it is necessary to move the level 
to continue the same upper contour lines, when (the staff" beiog 
placed at one of the pickets just driven) the vane is again raised 
or lowered to suit the next position of the axis of the instrument 
and kept at this height, as before, for the continuation of the 
line. To trace the next lower contour line, it is merely necessary 
to raise the vane on the staff, five, ten, or whatever number of 
feet may be the vertical distance determined upon, and proceed as 



115 

before. When the level itself has to be moved to lower ground, it 
must be so placed that its axis will cut the ground above one of the 
pickets of the line just marked out, and the same quantity of li?e 
or ten feet added to the reading of the staff at this picket, will 
give the height of the vane for the next lower horizontal line. 

" The use of driving all these pickets, marking out the contours 
nearly in the same line down the slopes, becomes evident when 
they are to be laid down on the plan, the places of the original 
banderoles or long pickets being fixed with reference to each 
other, it is only necessary to measure between them, entering the 
distances on these lines, with the offsets to the right or left to 
the different short pickets marking the horizontal lines." 

The instrument, best adapted for contouring where a rapid 
delineation of country is au object frequently of greater importance 
than accuracy, is the water-level its best recommendation being the 
facility with which it can be made and requiring no adjustment 
when using it. The following description is taken from " Erome 
on Surveying." 

The Erench water-level is much used, on the continent, in taking 
sections for military purposes. It possesses the great advantage of 
never requiring any adjustment, and does not cost the one-twentieth 
part of the price of a spirit-level. From having no telescope, it 
is impossible to take long sights with this instrument ; and it is 
not of course susceptible of very minute accuracy : but on the other 
hand, no gross errors can creep into the section, as may be the 
case with a badly adjusted spirit-level, or a theodolite used as such, 
the horizontal line being adjusted by nature without the interven- 
tion of any mechanical contrivance. As this species of level is not 
generally known in England, the following description is given ; 
which, with the assistance of the sketch, will enable any person to 
construct one for himself without further aid than that of common 
workmen to be found in every village. 

a 5 is a hollow tube of brass about half an inch in diameter, 



116 




and about three feet long, c and d are short pieces of brass tube 
of larger diameter, into which the long tube is soldered, and are for 
tlie purpose of receiving the two small bottles eand/, the ends of 
which, after the bottoms have been cut off, by tyiiig a piece of string 

round them when heated, 
are fixed in their positions 
with putty or white lead ; 
the projecting short axis 
g works (in the instru- 
meut from v\'hich the 
sketch was taken) in a 
hollow brass cylinder h, 
which forms the top of 
a stand used for observ- 
ing with a repeating cir- 
cle ; but it may be made 
in a variety of ways so as to revolve on any light portable stand. 
The tube, when required for use, is filled with water (coloured with 
lake or indigo), till it nearly reaches to the necks of the bottles, 
which are then corked for the convenience of carriage. On setting 
the stand tolerably level by the eye, these corks are both with- 
drawn (which must be done carefully and when the tube is nearly 
level, or the water will be ejected with violence) and the surface of 
the water in the bottles being necessarily on the same level, gives 
a horizontal line in whatever direction the tube is turned, by which 
the vane of the levelling-staff is adjusted. A slide could easily 
be attached to the outside of c and d, by which the intersection 
of two cross wires could be made to coincide with the surface of 
the water in each of the bottles ; or floats, with cross hairs made 
to rest on the surface of the fluid in each bottle, the accuracy of 
their intersection being proved by changing the floats from one 
bottle to the other ; either of these contrivances would render the 
instrument more accurate as to the determination of the horizontal 



117 

line of sight ; though cue of its great merits, quickuess of execu- 
tion, would be impaired by the first, and its simplicity affected by 
either of them. For detailed sections on rough ground, where the 
staff is set up at short distances ajpart, it is well qualified to super- 
sede the spirit-level ; and is particularly adapted to tracing contour 
lines. 



118 



COKEECTIONS FOR CURVATURE AND EEFRACTION. 

Showing the difference of the Apparent and True Level in Feet, and 
Decimal parts of Feet, for Distances in Feet, Chains, and Miles. 



Correction in 


Feet. 




Correction in 


Feet. 










'3 






r^ 














V 


a 


a; C 


r; 


<u 


C 


<u C 


S 


rH 


o 




o 


l-l 


o 
























-^ o 














03 


rt rt 


>M 


S3 


rt 


? c3 


• r-t 




















tM 










U V- -^ 






Pi 


3 ^ 


C3 




4; 

Pi 


13 aj 
UC3 




IH 


Ut 


>- -C3 


to 


S-i 


u 


t--d 


"ti 


o 


o 






o 


o 


o - 




^ 


(^ 


Ph Ti 


Q 


t^ 


fi^ 


fi- i 


p; 



Correction in Feet. 






100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

650 

700 

750 

800 

850 

900 

950 

1000 

1050 

1100 

1150 

1200 

1250 

1300 

1350 

1400 

1450 

1500 

1550 

1600 

1650 

1700 

1750 

1800 

1850 

1900 

1950 

2000 



•00024 

•00054 

•00096 

•00149 

•00215 

•00293 

■00383 

"00484 

•00598 

■00724 

'00861 

•01010 

•01172 

'01345 

'01531 

'01728 

"01938 

'02159 

'02392 

'02638 

'02895 

"03164 

'03445 

"03738 

"04043 

"04361 

"04689 

"05030 

"05383 

"05748 

'06125 

'06514 

"06914 

"07327 

"07752 

"08188 

"08637 

"09098 

'09570 



00004 
00008 
00013 
00021 
00031 
■00042 
00055 
■00069 
•00085 
■00103 
■00123 
•00144 
■00167 
•00192 
■00219 
•00247 
•00277 
•00308 
■00333 
•00377 
•00414 
■00452 
■00492 
'00534 
•00578 
•00623 
•00670 
•00719 
•00769 
•00821 
•00875 
■00931 
•00988 
•01047- 
•01107 
■01170 
01234 
01300 
•01367 



000-20 
00046 
•00083 
•00128 
■00184 
•00251 
•00328 
•00415 
•005 13 
•00621 
•00738 
•0OS66 
•01005 
•01153 
•01312 
•01481 
01661 
01851 
02059 
•02261 
•02481 
•02712 
•02953 
•03204 
•03465 
•03738 
•04019 
•04311 
•04614 
•04927 
■05250 
•05583 
•05926 
•0628(1 
•06645 
•07018 
•07403 
•07798 
•08203 



1-0 
1-5 
2 
2^5 
3^0 
3-5 
4-0 
4-5 
5-0 
5-5 
6-0 
6-5 
7^0 
7-5 
8^0 
8^5 
9-0 
9-5 
lO^O 
10^5 

iro 

11-5 
12-0 
12-5 
13-0 
13^5 
14^0 
14-5 
15^0 
15^5 
16-0 
16-5 
17^0 
17^5 
18^0 
18^5 
19^0 
19^5 
20-0 



00010 
00024 
Ci0042 
•00065 
•00094 
•00128 
■00167 
•00211 
■00261 
■00315 
•00375 
•00440 
00511 
•00586 
•006G7 
•00753 
•00844 
•00940 
•01042 
'01149 
•01261 
•01378 
•01501 
■01628 
■01761 
•01899 
•02043 
•02191 
•02345 
•02504 
•02668 
•02837 
•03012 
•03192 
•03377 
•03567 
•03762 
•03963 
•04169 



00001 
00003 
00006 
00009 
00013 
00018 
00024 
00030 
00037 
00045 
■00054 
■00063 
•00073 
•00084 
•00095 
•00108 
•00121 
•00134 
•00149 
•00164 
•00180 
•00197 
00214 
00233 
■00252 
•00271 
•00292 
•00313 
•00345 
•00358 
•00381 
•00405 
•00430 
•00456 
•00482 
•00509 
•00537 
•00566 
•00596 



00009 
■00021 
00036 
00056 
00081 
001 10 
■00143 
■00181 
•00224 
■00270 
■0032! 
•00377 
■00438 
•00502 
•00572 
•00645 
•00723 
•00806 
00893 
•00985 
•01081 
01181 
•01287 
•01395 
•01509 
•01628 
■01751 
•01878 
•02010 
•02146 
•02287 
•02432 
•02582 
•02736 
•02895 
■03058 
•03225 
•03397 
•03573 



1 

1| 
2 

2h 
3 

H 

4 

4| 
5 

5^ 
6 

H 

7 

71 
' 2 

8 

H 

9 

9h 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 



•0417 

•1668 

•3752 

•6670 

1 5008 

2^6680 

4-1688 

6^0030 

8-1708 

10-6720 

13^5468 

16-6750 

20^1769 

24^0120 

28^1809 

32^6830 

37^5190 

42-68S0 

48^1910 

54^0270 

60^1971 

66^7000 

80-7070 

96-0480 

112^7230 

130^7320 

1500750 

170-7520 

192 7630 

216^10S6 

246-7870 

266^8000 



•0060 

•0238 

•0536 

•0953 

•2144 

•3811 

•5955 

•8561 

M673 

1-5246 

1-9295 

2-3821 

2^8824 

3-4303 

4-0258 

4'6690 

5-3599 

6-0997 

6-8844 

7-7181 

8-5996 

9-5286 

11-5296 

13 7211 

16-1033 

18-6760 

21-4393 

24-3931 

27-5376 

30-8727 

34-3981 

38^1143 



•0357 

•1430 

•3216 

•5717 

1^2S64 

2-2869 

5733 

1469 

0035 

1474 

11-5773 

14-2929 

17^2945 

20^5817 

24^1551 

28-0143 

321591 

36-5883 

41-3066 

46^3089 

51-5975 

57-1714 

69-1774 

82-3269 

96-6197 

1 12-0560 

128-6357 

146-3589 

165-2254 

1 85^2359 

206-3889 

228-6857 



119 



CHAPTEE YII. 



E A I L W A T 



C r E Y E S . 



Curves aro not necessary for plain roads, but it is always a much 
neater method to join two straight portions of any road by a re- 
gular curve than bj a mere bent line ; and it is also to be borne 
in mind that it may b@ desirable to convert a common road iuto a 
railway, in which case considerable trouble and expense wiU be 
saved, if all the alterations of directions have been made of regular 
curves. 

Curves may be laid out in several ways, a few only of the most 
useful methods for ordinary practice are given here. 

The length of radius to be given to a curve is manifestly inde- 
terminate, for if AB, EC be two 
y'>^ portions of a line of road meeting 

in E, and it be required to unite 
them by a circular arc DE, we 
have by plane trigonometry, DB 
= DO tan. DOE, and since DOB 
is constant, DB varies aa DO, we 
may therefore either fix the point D and its corresponding point 
E in the other lines, and find the radius DO by the above equation ; 
or we may assume the length of the radius DO, and thus determine 
the length of the tangent DB, that is, the distance of D from the 
intersection E of the two lines. 




120 



lu practice the first method M'iil generally he necessary, for if 
any obstruction P, as a aycII or house, be situated near the line, 
the commencement of the arc must be taken so that P may be en- 
tirely within or without the curve. Again if there be no features 
of the ground to determine the question, as often happens in India, 
frcm the open nature of the country, a good method will be to fix 
the point D at some convenient distance from A the commence- 
ment of the road, with reference to the division of the road into 
miles, furlongs, &c., remembering however that the radius thus de- 
termined should never be less than i mile, and that ceteris parihtcs, 
the greater the radius the better the road. 

The number of cords must depend on the amount of curvature ; 
the fewer chords there are, the less trouble there will be in laying 
out the curve, while on the other hand they must not be so long as 
to have any sensible inclination to one another. The versed-sine 
of half the angle subtended by the chord at the centre, gives the 
greatest deviation of the chord from the arc ; and by finding this 
versed-sine for difi'erent numbers of chords, the Engineer will, 
generally, after two or three trials, be enabled to fix their number. 

METHODS OF LATIlN^a OUT THE CURVE. 

\st Method, — Centre invisible, no angular instrument required. 

Find the length of the sine of 1° to the 
^. Be given radius, and lay ofi" this distance 

in continuation on the straight line as 

.s^ from B to C, and from the point thus 

\ obtained, lay ofi" at right angles the 

\ versed-sine to the same augle and ra- 

dius. The first point D in the curve is 

thus fixed and by producing the chord 

BD (" 2 sin 30' X radius,) to a distance DE (= BD cos 1°,) and 

from the point E setting off at right angles EF = BD sin 1°, a 



121 




second point is tlius obtained. By continuing to produce tlie last 
chord in a similar manner and setting oil the offset, the remaining 
points are established. 

2ncl. Iletliod. — Same conditions as 1st. vSometimes the ground 

without the curve, only, is adapt- 
ed for chain measurem.ents. In 
which case, when the curve is 
not a long one i. e., does not ex- 
ceed say one-quarter of its ra- 
dins, the following method may 
be found useful. Lay off equal 
distances Ap^, p^ p^, p^ p^ &c., 
from A along AC^ and perpen- 
dicular offsets p>i ^'-1 !>■> ''^^2' ^^-f 
the points m^ m.^, &c., fixing the 
curve. 
To find these offsets. 

p)^ m^ = Ap^ tan. CAw?j 
2?^ m^ = 2^ X p^ m^ 
p^ mn — - n^ X p^ m^ 
Por draw m^ n^, m^ n^ &c., perpendicular to AO 
Then v.i^ n^ == Ap^ = An^ (2r—An^ 

= An^ X 2r nearly, since A??^ is verj^ small 
compared with r 

=^ p^ «?i X 2r 
w^ n."^ — Ap.^ cz: 2- Aj)^ = An^ X "Ir nearly 
= p^ m^ X 2r 

or p^ m^ =^ ^2? p^ m^ 
similarl}^ p^ m^ =z S^ 2\ '^i 

&C. =. &G. 

Case. II. Jhj ilie same method to lay out ilie curve when it is a 
long one. In a long curve, the tangents, if prolonged to their point 



122 




of meeting, would necessarily fall at a great distance from the curve, 
thus giving an incon- 
venient length to the 
offsets which in prac- 
tice should never ex- 
ceed two chains. To 
remedy this incon- 
venience, the curve 
must be divided into 
two or more parts, 
by introducing one 
or more additional 

tangents, and thus the offsets may be confined within their proper 
limits. In the annexed figure the curve AC is divided into two 
unequal parts at B ; at which point the tangent DBE is introduc- 
ed to meet the tangents AD, CE in D and E. The tangent AD 
must first be measured to an extent not exceeding one-eighth of the 
radius AO. Then in the right-angled triangle ADO, AO, DO are 
given from which the angle ADO can be formed, this angle, being 
doubled, gives the angle ADB which determines the direction of 
the tangent DBE. If no angular instrument be at hand to lay ofi:' 
this angle, the length of CE can be calculated and by measuring 
off the distance CE, the point E will be obtained, and by joining 
D and E, the tangent DBE. This done, the ofi:sets to the curve 
may be laid ofi" as in the last case the order of offsets being inver- 
ted in DB and again in BE. 

drd. Method. — Sa)ne conditions as first. Sometimes it may be 
most convenient to lay out the curve by oft'scts from its chord or 
chords, where obstructions, on its convex side, prevent the use of 
the proceeding method. Let ACB, be a portion, or the whole, of a 
railway curve ; HA, a tangent at its commencement ; TC, a tan- 
gent to its middle point C, Take if possible the chord AB, an 
even number of chains ; find the successive ofisets to the radius 



123 




AO and tlie tangent TC (r- AD = one-half AB,) the last offset 
TA will be = CD; from 
CD subtract the suc- 
cessive offsets, and the 
remainders will be the 
offsets p^ <2'i <^c. which 
must be set off in an 
inverted order from A 
to D, and their order 
must be again inverted 
in setting them off from 

D to B. If the curve be not yet completed, the operation may 
be continued hj taking other chords, as BE.* 

^tJi. Metliod. — Centre visible and accessible, -no angular instrument 
required. This method may sometimes be used with advantage 
especially for curves of large radii, find the centre of the curve by 
the intersections of the perpendiculars to the tangents at the 
points of contact, place a signal staff at that intersection, and also 
at the intersection of the straight lines produced. Divide these 
lines into any convenient number of equal parts, and set off at each 
in the direction of the centre a distance — \Jr^ + d'^—r, r being 
the radius of the curve, and d the distance from the point of con- 
tact to the point on the straight line. 

6tJi. Metliod. — Centre visible and accessible, angular insttn^nenf 
necessary. Where the curve is quick and the ground over which 
it passes is hilly, but at the same time is commanded from the 
points cf contact and centre, this method is attended with great 
advantages, in as much as the points are established independently 
of each other, and are free from all error which may arise (and 
which in the previous method must be allowed for) from the slop- 



• jVo/e.— It will be seen that the tangent TC, is not used in the operation further 
than to explain the nature of the method of obtaining the offsets. 

R % 



121. 



ing of the ground. This and the succeeding two are the simplest 
methods of all, as they require no calculation or measurement of 
offsets. They depend on the well known properties of the circle, 
that the angle contained by a tangent and a chord is equal to the 
angle in the alternate segment, and the angle at the centre is 
double of that at the circumference. The appropriate radius 
having been selected, the angle which a chord of about the length 
which it is required to have the points apart, would subtend at the 

circumference, (^^ e. the angle in the alternate segment) is computed 

. , „ , . f, ,.. ■ -, d X rad. of tables , 
irom the lormula, sine oi the angle :== , where. « 

2 r 

= length of chord, and r = radius of curve : then supposing r = 20 
chains, the angle which a chord of 100 feet would subtend would 
be 2° 10' 15'' ; with an ordinary theodolite this angle could only be 
laid off approximately ; an angle of 2^ would therefore be adopted 
the points due to it being 92-13 feet apart. This being chosen 
then as the angle, two theodolites would be set up one at B the 
other at 0, that at B ha- 
ving its telescope directed ^^ .^ 
on D, and that at O on B 
and having clamped their 
lower plates, the points in 
the curve wnll be obtained 
by the intersection of the 
arcs formed by moving 
them though 2^ and 4° re- 
spectively ; for the angle 
DB p == angle in alter- 
nate segment of the circle =: one-half the angle at the centre BOj?. 
An assistant must of course move by signal a flagstaff J until it is 
intersected by both theodolites when he will put down a picket to 
finally mark the spot. 

6th. Metliod. — Centre invisible, angular instrument required. 
The pcints in the curve in this method are fixed by intersections 




125 



from the points of contact of the tangents ; first find the angle 
ACB, suppose it to be 2a, then AOC = 90" — a, and if the re- 
quired curve is to be composed of n chords, and Ap^ is the first of 
these, Pi ^2 ^^® second, &c., 

Then Z. CA;^, =-. ^^""""^ 

n 

and /. CBp, = (90" -a) ^-^ 




of therefore a flagstafl" be moved until it comes into the intersec- 
tion ^^^ of Ajt^ and By^, ^^ will be a point in the curve. 

Similarly Z. Cj\. j^o =: 2 

Z CB«, = (90-a)'^^ 

and so on. 

7tJi. Metliod.—Same conditions as sixtli,~lt will often happen 
that the services of a second person capable of using the theodo- 
lite are not available, in which case the above method cannot be 
applied, the curve must tben be laid out by theodolite and chain, 
calculate the lengths of the chord Ap^, Ap.^, Ap.^, AF, (see last fig.) 
lay off banderoles along the lines Ax^, Ax^ &c. (the angles CA:rp 
QAx^ &c. being successively /3, 2 /3, 3 /3 &c. and /3 being found as 
already explained) and measure the chords along them. Proceed 
similarly from the point B. The point E the centre of the curve 



126 



slioulcl of course coincide with tlie same point as measured from A; 
if any small difference exist, the mean of the two points should be 
taken. To find the lengths of chords 

^ • A0», 
A^i = 2 sin — ^e^ 

A^2 = 2 sin AO/>j 

&C. r= &C. 

This method has the disadvantage of requiring a seperate cal- 
culation for each chord, but this will not be of consequence when 
the number is small. It posessess also the farther disadvantage 
that an error made in measuring the length of any chord will not 
be detected ; on the other hand, any error in one chord will not 
affect the accuracy of the rest, as would be the case in many of the 
methods ordinarily employed. 

Tlie Compound Curve, consists of two, three or more portions of 
arcs of different radii, and is adopted where the line is required to 
pass through given points to avoid obstructions, or where a princi- 
pal station or terminus is required. 

Case I. Tojiiid the radius of the compound curve, the starting point 
and one radius being given. 
From the given point B in 
the tangent AB, draw the 
given radius BO perpendicu- 
lar to AB ; and draw the curve 
to some point C, w^iere it is 
found convenient to change 
the radius ; draw the radius 
OC, and perpendicular there- 
to draw CT', meeting the tan- 
gent DT in T'; make TC 
= TC, and from C draw 
C\y at right angles to TC, 

meeting CO, prolonged if necessary, in 0' ; then 0' is the centre of 
the arc CC of the curve, conformable to the nature of tangents. 




127 



Case II. One of tlie two radii of the_ compound curve, and its 
sta7'ting and closing points heing given, to find tJie other radius. 

Let AB, CD be the tangents, B and Q' the starting and closing 
points of the curve. Draw the perpendiculars BO = C'H = given 
radius, to the tangents ; joiu OH, and bisect it in E; draw EO' per- 
pendicular to OH, meeting C'H prolonged in O' ; join 00' and pro- 
long it till OC ==: C'H: then the points 0,0', are the centres of 
the arcs BC, CC, which constitute the curve, O'C = O'C being 
the radius required. 

The Serpentine Curve, is used in railways when obstructions or 
some other cause' render its adoption preferable ; it consists of two 
circular arcs of different or the same radii, having their convex sides 
turned in opposite directions, like the letter S, whence it is some- 
times called the S curv^e ; the two portions of the curve have a com- 
mon normal at their point of junction, and therefore a common tan- 
gent at the same point. This curve affords the most easy means of 
joining two parallel or nearly parallel, portions of a line of railway. 

Case I. One radius and its tangential point heing given, to find 
the other radius and tangential point of the Serpentine Curve. 

Erom the given tangential point C draw the radius CO perpen- 
dicular to the tangent CD, and draw the curve CGr to some point 
Gr Avhere it is found convenient that it should have its point of 




.contrary flexure; throuj;h OGr draw the normal OGO'; from G 
draw GT at right angles to OGO' to meet the tangent AT; make 



128 



TB = TGr; aiid draw BO' perpendicular to AT, meeting OGO' in 
O' ; then O' is the centre, and O'B = OG- is the radius of the 
curve BGr, as is evident from the nature of tangents. 

Case II, When the tangential points and o?ie of the radii are 
given to find the other radius. 

From the given tangential points C and B draw CO, BH respec- 
tively perpendicular to the tangents CD and BA, and equal to the 
given radius ; join OH and bisect it in P'; draAV FO' at right angles 
to OH, jmeeting HB prolonged in O', and join 00' ; meeting O'Gr 
=: O'B, then is the centre, and O'B =: O'Gr is the radius of the 
portion BGr of the curve, as required. 

Case III. When the two portions have the same radius, to deter- 
mine that radius, the tangential points and their distance heing given. 

Let AB, CD be the tangents. B and C the given tangential points, 
and BC the given distance, draw Bo = Co' respectively perpendicu- 
lar to AB, CD and of any convenient length ; through o, parallel 
to BC draw o q indefinitely ; with the compasses apply o'o" = 2 Co' 



-"^0 




— 2Bo ; through C,o" draw Co" O, meeting BO prolonged in O ; 
and though O, parallel to o"o' , draw 00', meeting Co' prolonged 
in O'; then 00' are the centre, and OB and O'C are the equal 
radii of the serpentine curve BGC, the common normal of the por- 
tions Ba, GC of the curve, beiug OGO' rzr 2B0 = 2C0'. 

Curve of Deviation. In some cases it may be necessary to make 



129 



a given deviation from a straight line of railway, so that the works 
may avoid a building or other obstruction situated on or near it, 
this is done by means of three curves as follows. Let ABCD be a 
straight portion of the rail\^'ay, Ji a building or other obstruction 
on the line. Take HQ of a sufiicient length for a deviation, that 
the line may avoid the object at h ; and tiirough Q draw a curve 
GQGr' of radius QO' eq^ual to, or greater than one mile. Draw 
also two curves BGr, Gr'C, of like radius, to the first curve at Gr- 
and Gr', and the line at B and C ; then the lines 00' and O'O"' 

joining the centres of the 
curves, will pass through 
their contrary points of 
flexure at Gr and Gr'. Put 
r = common radius OB 
= O' Q = O'T, and d 
= required deviation = 




HQ ; then BH = H C ~ V*^ C^ ^ — d) and the four equal chorda 
BG-, CG-, &c., are each equal to s/d r. 

Having given these various methods of determining the radii and 
common normals, indicating the positions of the tangent pointa 
of the parts of the Compound, Serpentine and Deviation curves, the 
manner of laying out the curves themselves by the previous me- 
thods, according to circumstances, will be readily seen, recollect- 
ing, that when junction point of curves of different radii occur ; to 
commence the operation afresh, by using the radii and tangent of 
the respective portions of the curve. 



USEFUL PROBLEMS IN SURVEYING. 
Problem I. 
To draw upon the ground a straiglit line through two given points. 
Plant a picket, or staff, at each of the given points, then fix an- 
other between them, in such a manner that when the eye is plac-ed 

s 



130 

at tlie edge of one staff, the edges of the other two may coincide 
with it. The line may then be prolonged by fixing np other staves. 
The accuracy of this operation depends grciitly on fixing the staves 
upright, and not letting the eye be too near the staff from ^vhence 
the observation is made. 

Peob. II. 
To walJc in a straight line from a proposed ^oint to a given ohject. 
Fix upon some point, as a bush, or a stone, or any mark that 
you mtiy find to be in a line with your given object, and walk 
forward, keeping the two objects strictly in line, selecting a fresh, 
mark when you come within 20 or 30 paces of the one upon which 
YOU have been inovino^. Observe — that to walk in a direct line, 
it is always necessary to have two objects constantly in view. 

Peob. III. 
To trace a line in tlie direction of two distant points. 

Let two persons separate to about 50 or 60 paces ; then, by 
alternately motioning each other to move right or left, they soon 
get exactly into line with the distant objects : or, for greater accu- 
racy, they may hold up staves. 

In sketching ground, it is constantly necessary to get in lino 
between two objects : if these are not very distant, a well-drilled 
soldier can alway do so within a few paces (near enough for 
sketching purposes) by fronting one object exactly, and then 
fiicing to the right about ; when, if he finds himself accurately 
fronting the other object, he will be tolerably well in line with 
them. . 

A right angle may also be formed very nearly by fronting an 
object, and then facing to the rigid or hft. 

Peob. IV. 

IIoio to lag off a perpendicular loith the chain. 
Suppose A the point at which it is required to erect a right angle 



131 



fix an arrow iuto the ground at A, througli the ring of the. chain, 
marking twenty links ; measure 
foi^fi/ links on tlie line AB, and 
pin down the end of the chain firm- 
ly at that spot, then draw out the 
remaining eighty links as far as the 
chain will stretch, holding by the 
centre fifty -link brass ring as at C 
— the sides of the triangle are then 

in the proportion of three, four, and five, and consequently CAB 
must be a right-angle. 

An angle equal to any other angle can also be marked on the 
ground, with the chain only, by measuring equal distances on the 
sides containing it, and then taking the length of the chord : the 
same distances, or aliquot parts thereof, will of course measure the 




same angle. 



Peob. 




lo avoid an olstacle, sucli as a house, in your chain line. 

The usual way of avoiding 
an obstacle of only a chain 
or two in len^^th such as a 
house, is by turning off to the right or left at right angles till it is 
passed, and then returning in 
the same manner to the origi- 
nal line. 

A more convenient method 
is to measure on a line making 

an angle of 60^ with the ori- 

c 
gmal du'ection a distance suf- 
ficient to clear the obstacle, and to return to the line at the 
same angle, making CD = BC the distance BD is then equal to 
either of these measured lines. 




132 




Pros. YI. 

To find the length of the line AB accessible only at loth ends. 

Having fixed on some convenient 
point O, measure BO and AO ; and 
prolong those lines till OG — OB, 
and OD == OA ; then the distance 
between the points D and C ^Yill be 
equal to AB, for the sides of the 
triangle COD, BOA, about the equal 
angles at are respectively equal, 
therefore the third sides CD, BA, will also be equal. 

Peob. Yll. 

To find the distance of an inaccessible object by means of a 

rhombus. 

With a line or measuring tape, Avhose length is equal to the 
side of the intended rhombus, lay down 
one side BA in the direction BO, and 
let BC another side be in any conveni- 
ent direction : fasten two ends of two 
of those lines at G and A; then the 
other ends (at D) being kept together, 
and the lines stretched on the ground, 
those lines AD, CD, will form the other 
two sides of the rhombus. Set up a 
mark at E, where OG, AD, intersect; 
and measure ED ; then the sides of the 
triangles BDC, CBO, being respective- 
ly parallel, the triangles will be similar : hence, ED 

BO. 

Suppose the side of the rhombus is 100 feet, and ED = 11 ft, 7 

in.— then, ll/y : 100 : : 100 : 863 feet nearly = BO. 




DC :: CB 



133 



If be ground be nearly level, a rhombus, whose side is 100 feet, 
will determine distances to the extent of 300 yards within a very 
few of the truth. 

Peoe. VIII. 




To find flie length of the line AD, inaccessible at the point D» 
The measurement of the line AD, supposed to be run for the de- 
termination of a boundary, is stopped at B by a river or other 
obstacle. 

The point E is taken 
up in the line at about 
the estimated breadth 
of the obstacle from B ; 
and a mark set up at 
E at right angles to 
AD from the point B, 
and about the same 
distance as BE. The 

theodolite being adjusted at E, the angle EEC is made equal to 
BEE, and a mark put up at C in the line AD ; EC is then evident- 
ly equal to the measured distance EB. 

If the required termination of the line should be at any point C, 
its distance from B can be determined by merely reversing the 
order of the operation, and making the angle BEE' equal to BEC, 
the distance BE' being subsequently measured. There is no occa- 
sion in either case to read the angles. The instrument being level- 
led and clamped at zero, or any other marked division of the limb, 
is set on B : the z^^j^cr plate is then undamped, and the telescope, 
pointed at E, Avhen being again clamped, it is a second time made 
to bisect B ; releasing the plate, the telescope is moved towards D 
till the vernier indicates zero, or whatever number of degrees it 
was first adjusted to and the mark at C has then only to bo placed 



134 



in the line AD, and bisected hy tlie intersection of the cross wires 
of the telescope. 

If it is impossible to measure a right angle at B, from 
some local obstruction, lay off arj convenient angle ABE and 
set up the theodolite at E. 

Make the angle EEC equal to one half of ABE, and a 
mark being set up at C in 
the prolongation of AB, 
BC is evidently equal to 
BE, which must be mea- 
sured, and which may at 
the same time be made 
subservient to the purpose 
of dQlineating the boun- 
dary of the river. 




PnoB. IX. 

To find ilie distance to any inaccessihJe looinf, on tlie oilier side of a 
river, ivithout the use of any instrument to measure ancdes. 
Prolong AB to any point 
D; making BC equal to CD; 
lay off the same distances in 
any direction Dc — c5 : — mark 
the intersection E of the lines 
joining Be and Ql : mark also 
E the intersection of DE pro- 
duced, and of A5— produce Db, 
and BE, till they meet in a and 





135 



Peob. X. 

To find the ^oint of intersection of two Ihies meeting in a lal<:e or 
river, and tlie distance DIB to the point of meeting. 
Erom any point E on tlie line AX draw FD, and from any 
other point E draw ED, produce both these lines to H and G, 
making the prolongations either equal to the lines themselves, 
or any aliquot part of 
either length suppose 
one-half; join GIT, and 
produce it to O, where 
it meets the line CB, then 
OH is one-half of EB, 
and OD equal to half of 
DB ; which results give 
the point of intersection 
B, and the distance to it from D. 




PnoB. XL 

To fi'iid the hsiglit of a poi:it on an inaccessible hill ivithout the me 

of instruments. 
Drive a picket three or 
four feet long at H, and 
another at L, where the 
top of a long rod ED is in 
a line with the object S 
from the point A (the 
heads of these pickets be- 
ing on the same level) ; 
mark also the point C, 
where the head of the rod 
is in the same line with S, 
from the top of any other picket B, and measure AE and BC ; lay 




136 



off tlie distance BC from E to I, and the two triangles ADJ and 
ASB, are evidently similar, whence 



PS 



AB 

~Ab 



HI 



1 AP __ AB HI 

HO ^^^ AF —' A6~HO 



PS therefore 



DE. ^^ and AP 



AF.-^. 

HO 



Pbob. XII. 



To find your place in a Survey 

Let A and B be two sta- 
tions, whose places are fixed, 
and we want to determine 
the point C. Take the bear- i^ 
ing of A, 123° N.W. : having 
done which, we know, tliat 
C bears from A, 128° S.E. 
Adjust the protractor at A, 
by means of the east and 
west parallel lines, and lay 
off 128^ S.E. the bearing of 
C ; Avhich point C must, we 
know, lie somewhere in the 

line thus obtained. Next, take the beariTig of B 63° N.E., and 
having adjusted the protractor at B, lay of 63' S.AV., and where 
a line drawn from B (to represent this bearing) cuts the line or 
bearing drawn from A, is the required station C. 

The above may be put into short rule : thus — To find your sfa- 
Hon hy ohservations talcen to two j^oints already known. Protract 




Note. That the nearer your tno bearings meet at a right angle, tlie more 
correct will the station be determined: and also, that when a third fixed point can 
be seen, a bearing to it will serve to corrohoraie your other observations; and a 
point so obtained, namely, by the exact meeting of three bearings, becomes as good 
as any other point. 

Tlie above is a very useful problem iudeed, indispensable when sketching 
ground and filling in a survey. 



137 




from those points the opposite bearings to what you observe, and 
their intersection fixes the place sought. For example, if the bear- 
ing to a point be 20° N.E., protract from that point 20° SAY., &e. 

Peoe. XIII. 

To reduce the offset-piece ABODE to a riglit angled triangle 
AEc, hy an equalizing line Ec, loith the parallel ruler. 

Draw the indefinite line Ac nerpendicular to AE. Lav the 

parallel ruler from A to C ; 
hold the near side of the ruler 
firmly, and move the further 
side to B, which will cut Ac? 
at a^ where a mark must be 
made. Lay the ruler from a 
to D, and the further side 
thereof being now held fast, 
bring the near side to C, marking A<? at h. Lay the ruler from h 
to E, move it parallel, to D, marking Kc at c. Join Ee ; then 
AEc? is the right angled triangle required, and its area may be 
found by taking half the product of AE and Ac. 

THE FOLLOWING IS A GEJfEBAL EULE EOE SOLVIIS'G PROBLEMS OF 

THIS Ki:!?©. 

Draw a temporary line as Ac at right angles, or at any other 
angle to the chain line, as AE, of the offsets. 

1. Lav the ruler from the first to the third angle, and move it 
parallel to the second angle ; then make the first mark on the 
temporary line. 

2. Lay the ruler from the first mark on the temporary line to 
the fourth angle, and move it parallel to the third angle ; then 
make the second mark on the temporary line. 

3. Lay the ruler from the last named mark to the fifth angle, 

T 



138 



and move it parallel to the fourth angle ; then make the third mark 
on the temporary line. 

4. Lay the ruler from the last named third mark on the tem- 
porary line to the sixth angle, and move it parallel to the fifth 
angle ; then make the fourth mark on the temporary line. 

In this manner the work of casting by the parallel ruler may 
be conducted to any number of angles. Great care must be taken, 
during the operation, to prevent the ruler slipping, as such an ac- 
cident will derange the whole of the work, if not discovered and 
immediately corrected. 

Frob. XIV. 

To reduce a curved offset-piece to a right-angled triangle. 

Let Xabcde^ be the curved offset-piece. Divide the curve by 
points «, hy &c., so that the parts 
A«, al, &c., may be straight, or 
nearly so; and draw A5 perpen- 
dicular to AB. Lay the ruler 
from A to & ; move it parallel to «, 
and mark A5 at 1. Lay the ruler 

from 1 to c ; move it parallel to 5, and mark A5 at 2. Lay the 
ruler from 2to d; move it parallel to c, and mark A5 at 3. Lay 
the ruler from 3 to e ; move it parallel to d, and mark A5 at 4. 
Lay the ruler from 4 to B ; move it parallel to e, and mark A5 
at 5. Draw the line B5 ; then will AB5 be a right angled tri- 
angle equal in area to the offset-piece KalcdeB, as required. 

Prob. XV. 

To reduce the irregular -field ABCDEFGHK to a trapezium by 

the parallel ruler. 
Prolong the line AK at pleasure. Lay the ruler from K to G- ; 
move it parallel to H, and mark AK prolonged at 1. Lay the 




139 



ruier from 1 to F ; move it parallel to Gr, and mark AK at 2, 
Lay tlie ruler from 2 to E : move it parallel to F, and mark 
Al at 3. Draw a line 8 to E and prolong from E. Lay the 
ruler from E to C ; move it parallel to D, and mark 3E at 4. 
Lay the ruler from 4 to B ; move it parallel 
to C, and mark IE prolonged at 5. Draw 
a line from 5 to B ; then shall AB53 be 
a trapezium, equal in area to the irregular 
figure ABCDEFGrK ; the area of which may 
be found by multiplying the diagonal B3 by 
half the sum of the perpendiculars thereon 
froiiii A and 5. 

Note, fn this manner the crooked sides of a field 
may be successively reduced to straight ones. Thus, if 
the side AB had been crooked, the operation of straight- 
ening might be continued by prolonging the dotted line 
5B, and find successive points therein, corresponding 
to the assumed angles, till the last angle was brought 
thereon, and so on with respect to the side AK, had it 
3Jso been crooked. When the sides of a field are curved, 
the method of reducing them to straight lines is the same as sh \vn in Problem XIV. 

Peoe. XYII. 

To draw an equaTizinr] line tlivougli the crooJced fence abode, -to 
that the two fields ABea, aDCe may he four sided. 

Lay the ruler from a to 
c ; move it parallel to h, ^ 

and mark AD at 1. Lay 
the ruler from 1 to d\ move 
it parallel to c, and mark 
AD at 2. Lay the ruler 
from 2 to e ; move it par- 
allel to d ', and mark AD 
at 3. Draw the line e 3, 
and it will divide the two 





T I 




140 

fields, so that their quantities shall be the same as those before 
separated by the crooked fence abode. 

It is scarcely necessary to add, that had the fence ahcclc been 
curved, the equalizing line might have been found as in Prob. XIV. 

SCALES. 

As the representation of any object is usually less than the ob- 
^ ject itself, the ratio between the lineal dimen- 
sions of the former and those of the latter, con^ 
stitute what is called the " Scale of the Draw- 
ee \y ing" ; and this ratio should be expressed on the 
drawing, either by a numerical fraction expressing that ratio, or 
else by a line divided into equal parts each of which represent one 
or more units of length, by which the object itself is measured. 

Eor instance, let ABCD be the plan of a room, one side of 
which, AB, is 2|- inches, while the room itself is 40 feet ; then the 
former is on a scale of 2^ inches to 40 feet, or of 1 inch to 16 feet 
that is of 1 inch to 192 inches, and, since 192 inches are represent, 
ed by 1 inch, we have y^^ expressing the proportion that the lines 
of the drawing bear to those of the object and this is termed, 
" the representative fraction of the scale." 

By means of this fraction the dimensions of the object can be 
readily obtained in the linear measure of any country, by multi^ 
plying the length of the corresponding part in the drawing, referred 
to such measure, by 192, or generally by the denominator of such 
fraction. 

But in order that the true dimensions of the room may be at 
once ascertained from the drawing, without the necessity of per- 
forming for each a fresh calculation, it is requisite to construct its 
scale. 

To obtain this it is convenient that the whole length represented 
by the scale be some multiple of 10, and that the scale be suflSci- 



entij long for taking off at one measurement long lines, though 

not necessarily the longest lines of the drawing. 

Example. — In the present case let the scale shew a length of 50 

feet, then if cc be the required length in inches to represent 50 

feet, we have, 

1 a: 



192 50x12 
50x12 50 



cc = 



3-125 inches. 



192 10 

Draw three fine parallel lines in pencil at -—th. inch apart, and 
set off on the lowest line a distance of 3*125 inches. Now divide 
this distance into 5 equal parts, termed ^^nm^^r^ divisions, each of 
which will represent 10 feet. 

'(' 5 io 20 30 *oj-eei. 



The left hand one will further be subdivided into 10 equal 
parts, called seoondary divisions, each of which represents 1 foot. 
The two lower lines are to be drawn in ink, the lowest being the 
darker. The perpendiculars through the primary divisions are 
drawn to the top line, those through the secondary, to the second 
line only, except the centre one, which is carried up half way be^ 
tween the two upper lines. The line at the right hand end of the 
secondary divisions is marked O, (zero,) the primary divisions are 
numbered successively 10, 20, 30, &c., to the right, and the se- 
condary ones 5, 10, to the left of it. The word expressing the 
units of the measure, in this case " feet" ; is printed immediately 
after the last number, 40. 

Example. — Construct a scale of 11 yards to 1 inch to shew 60 
yards. 

Here 11 : 60 : *. 1 : a: 



X =r — -— == 5*45 inches. 



142 



or, if we proceed as in the preceding example, we find the repre- 

1 
sentative fraction of the scale, which is — — -— 

11x36 

^ 11X36 "~ 60x36 

60 
.*. ,r u= -— - = 5*45 inches. 

11 

Draw the three fine parallel lines and set off 5*45 inches ; in this 
case divide that distance into sLr equal parts and proceed as before, 
observing that the unit of measure is " yards." 

If the representative fraction of the scale be given, the unit of 
measure as well as the length of the required scale, should also be 
given. 

Eor 3^ represent a scale, of 11 yards to 1 inch. 
Ditto, do. 33 feet do. 

therefore a scale of yards or of feet fulfils the conditions. 

"When the representative fraction is very small, it is not neces- 
sary that the secondary division should be arranged as units ; they 
may, according to convenience, represent 10 or 100 units, but the 
decimal notation is to be retained, i. e., the primary divisions are 
to be each equal to 10 of the secondary divisions — this enables 
any length to be taken ofi" by inspection, and without calcula- 
tion. With certain units the left hand primary division cannot be 
divided into ten equal parts ; for instance, if miles and furlongs were 
to be shewn, the secondary division would be ei^lif in number ; if 
feet and inches, twelve, and so forth. 

The foregoing are termed " Simple" or " Plain Scales," in which 
only the primary division and its parts, termed the secondary divi- 
sions, are shewn ; but if it be necessary to obtain a more minute sub- 
division, by representing the parts of the secondary division them- 
selves, recourse must be had to Vernier or Diagonal scales. The 
former have already been described at the beginning of Chap. III. 
All that has been said there relative to arcs, is equally applicable 
to straight lines. 



143 

Diagonal Scales. — Construct a diagonal scale of oue-eightb, or of 
1*5 inches to 1 foot, to read to -j^q of a foot, shew 2 feet. 






'0 38165432') 
A C 



Draw AB three inches in length, bisect it in C, and subdivide 
the first primary division AC into 10 equal parts. Draw above 
and parallel to AB as many equivalent lines as are required ; in 
this case 10 ; for y^ = -^ of ^\ ; also draw AE, CD, BF, per- 
pendiculars from AB to E!F, and divide DE similarly to AC, 
joint the O (zero) point in AC with the first secondary division in 
DE, the first division in AC with the second in DE, and so forth. 

Print 2, 4, 6, 8, 10 upwards, at the left hand end of alternate 
horizontals. Since the triangles formed by CD and the line drawn 
from the O (zero) point in AB to the first secondary division in 
DE, intersecting the horizontal lines, are similar, we have the bases 
of those triangles representing successively yV> i^> ^^-j of a secon- 
dary, and consequently -j^, y^-^, &c., of a primary division. 

If it be necessary, with the same representative fraction -^th, 
to construct a scale of feet and inches, and to read diagonally 
to xoth of an inch, we divide AC into tivelve equal parts and 
proceed as before. 

Example. — Construct a scale of one mile to 4 inches in yards, 
and reading diagonally to 10 yards, to measure 2,000 yards. 

Here the representative fraction = — = -- 

^ 1,760x36 440x36 

1 X 2,000 50 



. . , X = .f!^^ = .f^L. =z 4-54 inches. 

440X36 2,000x36 440 11 

A length of 4*54 inches is divided into two equal parts, each repre- 
senting 1,000 yards, the left hand one is subdivided into 10 equal 



144 



parts, each representing 100 yards. Ten parallel lines and tlie 
diagonals, &c., as before, complete the scale to fulfil the conditions. 

Comparative Scales.-— Jn examining a drawing or map when the 
scale is constructed according to the linear measure of any Foreign 
country, it may often be necessary to determine the varions dis- 
tances in English, or other measure, in which case another scale is 
prepared, having an}?- convenient unit of measure. This second scale 
is termed "a comparative scale" to the first, and is very easily 
obtained. 

Example. — A Map of the Crimea is given, on which the scale is 
of E-ussian versts ; it is required to construct a comparative scale 
shewing English miles. 

Beferring to the scale of versts we observe that 50 versts meas- 
ure 6"35 inches. Hence as the verst = 1,166"6 yards, 

is the representative fraction of the scale, and 

50x1,166-6x36 ^ 

assuming that the required scale be sufficiently long to measure 30 

miles, proceeding as before wdth simple or plain scales — 

1 6"35 x 

we nave = 

50 X 1,166*6 X 36 30 X 1,760 x 36 

X = 6;35x^30 x_l^ ^ ^.,^^ .^^^^^ 

50x1,166-6 

The length of 5*748 inches is subdivided into three equal parts, 

each representing 10 miles, and the left hand primary division is 
subdivided into 10 equal parts, each of which is one mile. The scale 
is completed as before. Each of these scales having the same re- 
presentative fraction a-soVsT) ^^^ correctness of the comparative 
scale cannot be doubted. 

ON COPYING PLANS, MAPS, &C. 

Theee are several methods of doing this when the copy is to be 
of the same size as the original, such as placing the plan to be copied 



145 

with a sheet of paper over it on a tracing glass, placed in such a 
position that a strong light may fall on it from behind, and then 
tracing it off, or by placing a sheet of thin paper, having its under 
side blacked (by rubbing finely powdered black lead, or soft lead 
pencil over it,) on the sheet of paper that is to receive the copy, the 
original^ being placed over both, and the whole made steady by pla- 
cing weights thereon. All the lines of the copy must how be careful- 
ly passed over with a fine tracing point, and with a pressure propor- 
tionate to the thickness of the paper. The paper beneath will re- 
ceive corresponding marks, forming an exact copy, which may 
afterwards be inked in. "When the drawing is to be reduced or 
enlarged, the pentagraph or the method of copying by squares must 
be resorted to. 

The Pentagraph consists of four rulers, AB, AC, DE, and 'EF, 
made of stout brass. The two longer rulers, AB, and AC, are 
connected together at A, and have a motion round it as a centre. 
The two shorter rulers are connected in like manner with each 
other at F, and with the longer rulers atD and E, and, being equal 
in length to the portions AD and AE of the longer rulers, form 
with them an accurate parallelogram, ADEE, in every position of 
the instrument. Several ivory castors support the instrument, par- 
allel to the paper, and allow it to move freely over it in all direc- 
tions. The arms, AB and DP, are graduated and marked J J, &c., 
and have each a sliding index, which can be fixed at any of the divi- 
sions by a milled-headed clamping screw, seen in the engraving. 
The sliding indices have each of them a tube, adapted either to 
slide on a pin rising from a heavy circular weight called the fulc- 
rum, or to receive a sliding holder with a pencil or pen, or blunt 
tracing point, as may be required. 

When the instrument is correctly set ; the tracing point, pencil, 
and fulcrum will be in one straight line, as shown by the dotted 
line in the figure. The motions of the tracing point and pencil are 

U 



14G 



then, each compounded of two circular motiona, one about the 
fulcrum, and the other about the joints at the ends of the rulers 
upon which they are respectively placed. Ihe radii of these 
motions form sides about equal angles of two similar triangles, of 
which the straight Hue BC, passing through the tracing point, 
pencil, and fulcrum, forms the third side. The distances passed 
over by the tracing point and pencil, in consequence of either of 
these motions, have then the same ratio, and, therefore, the dis- 
tances passed over, in consequence of the combination of the two 
motions, have also the same ratio, which is that indicated by the 
setting of the instrument. 




vi:®* 



Our diagram represents the pentagraph in the act of reducing a 
plan to a scale of half the original. For this purpose the sliding 
indices are first clamped at the divisions upon the marks marked |; 
the tracing point is then fixed in a socket at C, over the original 
drawing ; the pencil is next placed in the tube of the sliding index 
upon the ruler DF, over the paper to receive the copy ; and the 
fulcrum is fixed to that at B, upon the ruler AB. The instrument 
being now ready for use, if the tracing point at C be passed deb- 



U7 



cately and steadily over every Hue of the plan ; a true copy, but 
of one-half the scale of the original, will be marked by the pencil 
on the paper beneath it. The fine thread represented as passing 
from the pencil quite round the instrument to the tracing point at 
C, enables the draughtsman at the tracing point to raise the pencil 
from the paper, whilst he passes the tracer from one part of the 
original to another, and thus to prevent false lines from being 
made on the copy. The pencil holder is surmounted by a cup, 
into - which sand or shot may be put, to press the pencil more 

heavily on the paper, when found necessary. 

If the object were to enlarge the drawing to double its scale, 
then the tracer must be placed upon the arm DF, and the pencil 
at C ; and if a copy were required, of the same scale as the original, 
then, the sliding indices still remaining at the same divisions upon 
D¥, and AB, the fulcrum must take the middle station, and the 
pencil and tracing point those on the exterior arms, AT5 and AC, 
of the instrument. 

Let figure 1, in the annexed engraving represent a plan of an 
estate, which it is required to copy upon a reduced scale of one- 
half. The copy will therefore be half the length and half the 
breadth, and consequently will occupy but one-fourth the space of 
the original. 



JF{^. 3. 



Fls. 2. 





m, 



u 2 



us 

Draw the lines PI, FGr at right angles to each other ; from the 
point r towards I and Gr, set off any number of equal parts, as Ya, 
a I, 1) c, &c., on the line E I, and F i, i Jc, k I, &c., on the line F 
Gr : froui the points in the line F I, draw lines parallel to the other 
line FGr, as a a, h h, c c, &q., and from the points on FGr, draw 
lines parallel to FI, as i i, k 7c, I I, &c., which being sufficiently 
extended towards I and Gr, the whole of the original drawing will 
be covered witli a net- work of small but equal squares. Next draw 
upon the paper intended for the copy, a similar set of squares, but 
having each side only one-half the length of the former, as is repre- 
sented in figure 2. It will now be evident that if the lines AB, 
BO, CD, &c., figure 1, be drawn in the corresponding squares in 
figure 2, a correct copy of the original will be produced, and of 
half the original scale. Commencing then at A, observe where in 
the original the angle A falls, which is towards the bottom of the 
square, marked d e. In the corresponding square, therefore, cf the 
copy, and in the same proportion towards the left hand side of 
it, place the same point in the copy : from thence tracing where 
the curved line AF crosses the bottom line of that square, which 
crossing is about two-fifths of the width of the square from the left 
hand corner towards the right, and cross it similarly in the copy. 
Again, as it crosses the right hand bottom corner in the second 
square below d e, describe it so in the copy : find the position of 
the points similarly where it crosses the lines ff and ^' (/, above 
the line I Z, by comparing the distances of such crossings from 
the nearest corner of a square in the original, and similarly marking 
the required crossings on the corresponding lines on the copy. 
Lastly, determine the place of the point B, in the third square 
below (J h on the top line ; and a line drawn from A in the copy, 
through these several points to B, will be a correct reduced copy of 
the original line. Proceed in like manner with every other line oi^ 



149 

the plan, and its various details, and you will have the plot or 
drawing, laid down to a small scale, j^et bearing all the proportions 
in itself exactly as the original. 

It may appear almost superfluous to remark, that the process of 
enlarging drawings, by means of squares, is a similar operation to 
the above, excepting that the points are to be determined on the 
smaller squares of the original, and transferred to the larger squares 
of the copy. The process of enlarging, under any circumstances^ 
does not, however, admit of the same accuracy as reducing. 



150 



CHAPTEE VIII. 



TEiaONOMETBICAL SUEYEYING. 

The basis of an accurate survey must necessarily be an extended 
system of Triangulatlon, the preliminary step in wbicb is the careful- 
measurement of a Base Line on some level plain : at eacb extremity 
of this base, the angles are observed between several surrounding 
objects previously fixed upon as Trigonometrical Stations ; and also 
those subtended at each of these points by the base itself. The 
distances of these stations from the end of the base line and from 
each other are then calculated, and laid down upon paper, forming 
so many fresh bases from whence other Trigonometrical points are 
determined, until the entire tract of country to be surveyed is 
covered over with a net-work of triangles, of as large a size as is 
proportional to the contemplated extent of the survey, and the 
quality and power of the instruments employed. The interior 
detail between these points is filled up either by measurement with 
the chain and theodolite, or by partial measurement, (principally 
of the roads,) and by sketching the remainder with the assistance 
of some portable instrument. 

For the description of the regular Trigonometrical Survey of a 
country, the reader must refer to larger works on the subject. 
What will be described here, is such a survey as might be made 



151 

with a 5-iiicli theodolite, if the surveyor had some few square miles 
of country to survey accurately. 

In fixing upon an appropriate site for the measurement of a Base 
Line, a level piece of ground should obviously be selected, v^here 
both ends of the base v^^ould be visible from the nearest trigonome- 
trical points. It should also be as near the centre of the survey as 
possible, but this is not absolutely necessary. For a survey of the 
extent abovementioned, it should be about 2,000 feet long, and the 
sides of the triangles may be increased to a mile or more. The pro- 
cess of measuring the base line is as follows: — The theodolite being 
set up at one end, the inclination of the ground, as far as it con- 
tinues the same, is measured by sending a staft' with the vane set to 
the height of the instrument, to the point where the change of 
inclination takes place, the distance between these points is then 
measured carefully with the chain, both forwards and backwards. 
The chain must be compared with a standard bofore measuring the 
base line, and afterwards, and the mean of the measurements taken 
for its true length. The theodolite is then removed to the place 
where the staff was held, which is called the second station, and the 
angle of elevation of the portion of the base line already measured 
re-taken, as well as that of the next portion, up to the 3rd station. 
The distance between the 2nd and 3rd stations is then measured 
twice as before, and in this way the whole length of the base line is 
measured, and also the inclinations of the ground with a view to re- 
ducing it to one horizontal line. Form A, at the end of the Chapter, 
shews the method of entering these observations in the Field-book. 
The trigonometrical stations must be chosen with a view to the 
formation of loell-conditloned triangles, i. e. triangles none of whose 
angles are less than 30° ; the nearer the triangle approaches to the 
equilateral the better. The sides of the triangles should increase 
as rapidly as possible from the measured base. The accompanying 
sketch shews how this is to be managed without admitting any ill- 
conditioned triangles. 



152 



AB is supposed to be the measured base, and C and D the 
nearest Trigonometrical points. 
All the angles being observed 
and the length of AB having 
heen measured, the other sides 
of the triangles DAB, CAB may 
be calculated. We can then 
calculate DC from the two tri- 
angles DAC, DBC, (having the 
two sides and included angle of 
each given,) one calculation act- 
ing as a check upon the accu- 
racy of the other. This line 
DC is again made the base from 
which the distances of the tri- 
gonometrical stations E and E 
are computed from D and C, and 

these lines ED, EC, DP, CE, can be used as fresh bases for ex- 
tending the triangulation, or if these be not sufficiently large, the 
length of EP can be calculated and used as a base. This is the 
usual method of starting from a base line, unless the nature of 
the ground to be surveyed, interferes. 

The remainder of the trigonometrical stations must be arranged 
over the whole survey, as the nature of the country will best allow, 
and care must be taken that no point in the survey is too far from 
some one of these stations. 

The best form of signal for a station is a basket, covered with 
canvas and white-washed over, fixed on the top of a lamhoo. The 
observations being made to the bamboo immediately under the 
basket, or if the stations be very far from one another, to the centre 
of the basket. These should be fixed in trees, or if there are no trees 
near the spot where it is wished to have a station, the bamboo 




153 

should be lashed to a hdlee, and the end of the latter sunk a foot 
or two in the ground, and the whole made firm and steady with 
ropes pegged into the ground, after the manner of a flag staff. 

Having selected all the stations and placed the signals, all the 
angles of the triangles must be observed with the theodolite, and 
for obtaining the relative altitudes of the ground at the different 
stations, the vertical angles also. The angles are observed from 
each station in succession, as follows : — 

The theodolite is centered over the point on the ground, marking 
what is called the station dot, which is vertically under the signal.* 

It is then levelled, and the A vernier being set to 360°, or zero, 
the intersection of the wires is made to bisect one of the stations, 
by turning the whole instrument round, the bottom plate is then 
clamped ; the vernier plate is now loosed and each signal that is 
visible is intersected in turn, and the horizontal and vertical angles 
read off and entered in the field-book (see form Bj. In the hori- 
zontal angles, the columns A,"B,C, are the minutes read off from the 
three verniers, the degrees being all read from the same vernier. 
In the vertical angles, columns A and B are the minutes read ofl: 
on the two arcs of the Everest theodolite ; if a T theodolite be used, 
there will only be one column. Having completed the circuit, and 
on re-intersecting the first station, the vernier should of course read 
360° ; if it does not, the under plate could not have been properly 
clamped, or the whole instrument must have been moved, and as it 
is not possible to find where the error occurred, all the angles must 
be re-observed. To prevent the omission of this check, the first 
station is re-entered after all the others. The 2nd set of angles are 

* This is found in the following manner. — Set up the theodolite at a short dis- 
tance from the signal, and having levelled it, fix the intersection of the wires on the 
signal, clamp the lower plates and bring ilie telescope down till it intersects the 
ground a foot or so beyond the signal. Make a mark at this point, and stretch the 
chain from the theodolite to it. Now remove the theodolite to another point, so 
that the direction between it and the signal, is about a right angle with the last 
direction and repeat the operation ; the point where these two lines intersect will be 
vertically under the signal. 



154 

observed witli tlie face of the instrument reversed, by which means 
any error in collimation is eliminated, and the angles commence 
from 180° instead of 360°, tbis tends to eliminate the errors that 

must necessarily exist in the division of the arc, by taking the 
angles at different portions of it, and taking their means. If the 
instrument has only two verniers, three sets of angles should be 
taken.* 

The height of the theodolite, and of the signal, from the ground 
must be measured, as also the height of the station if the angles 
should be observed from any artificial elevation, such as the roof of 
a house. These data are necessary for calculating the altitudes. 

Sometimes church steeples, prominent buildings, or other mark- 
ed objects, are peculiarly adapted for trigonometrical stations, but 
they have this disadvantage that the theodolite can seldom be set 
up immediately over the point observed. "When such is the case, 
a station called a satellite is chosen, as near as possible to the prin- 
cipal station, and the angles are taken from this point. It is clear 
that these angles before they can be employed in the computation 
of the triangulation, will be required to be transferred to the princi- 
pal station, that is they must be reduced to what they would have 
been had they been observed from that point — this is technically 
called tJie reduction to tlie centre. The angles are entered in the 
field-book in a similar manner to that of the ordinary stations, the 
necessary extra data being the distance of the principal from the 
satellite station and its direction. 

The observation of the angles completes the field work portion 
of the trigonometrical part of the survey. The various calculations 
must now be made and entered in the calculation book. The Base 
Line is first reduced to one horizontal line (see Form C) and its 
true length obtained. The horizontal and vertical angles are then 
meaned out from the field book, and entered in the calculation book 
(see Form D). The angles taken from satellite stations must next 

• If the Instrument be a Y theodolite, the 2nd set must be taken with tlie 
telescope inverted. 



■■P 



IIIIIIIIIIIIIIIIIIIIICIIIIIII 



• I > I 



.53 



be reduced to the centre, and then we have all the data ready for 
computing the sides of the triangles and the relative heights of the 

stations. 

The principle on which the angles of satellite stations are re- 
duced to the centre is as follows :— Suppose P to be the principal, 
S the satellite station, ABC three other trigonometrical stations. 
The theodolite having been set up at S, and suppose, for conveni- 
ence of illustration, that when set at 360° it was directed on P, the 
readings of A, B and 
C are taken. Let PS 
he produced both ways 
to m and n. J^ow the 
direction of A from S 
is represented by the 
angle mSA, and the 
same referred to P, by 
the angle mPA, but the 
angle mVA = «^SA -[- 
SAP. If, therefore, we 
find the angle SAP, and add it to the angle mSA, we shall obtain 
the angle mPA or the direction of A from P, which was required. 
To find the angle PSA, the necessary correction, we have the angle 
PSA, the side SP, and we can obtain the side PA, which is 
only required approximately since it is so much larger than SP., 
either by construction or by calculation, by the latter most cor- 
rectly. In the case of the next station B, the angle mS'B (taken 
in the direction of the dotted circle) is the angle we have observed, 
and the one we require is the angle mPB ; and the latter is less 
than the former by the small angle at B.* In this case, therefore, 
we must find the angle B and subtract it from the reading of B, to 
obtain what it would have been, if observed from P instead of S. 

* For the /. hPB = ^ «SB — Z. B, and adding 180° to each /i rwPB 
= Z. wiSB — Z. B. 

X 2 




i5G 

We have, therefore, this rule that standing at the satellite and fa- 
cing the principal station, the corrections for all the angles in the 
Qnglit semicircle are additive, for those in the left semicircle the cor- 
rections are stiitractive. Form E, shews the way of reducing the 
angles taken from a satellite station. In the first part, the column 
headed ' observed angles' gives the angles as taken from the field- 
book ; the next two columns are for finding the angle PSA, PSB, &c., 
(by deducting the reading of the principal station from that of each 
trigonometrical station successively, or vice versa, as the case may 
require,) which are then used below for the computation of the cor- 
rections, viz. the small angles A,B,C, &c. These being found are 
entered above, and applied to the " observed angles," which gives 
the " corrected angles " shewn in the last column ; in the example 
given, all the angles lying in the left semicircles, the corrections are 
all subtractive. These corrected angles can now be used for the 
determination of the angles of the triangles, as the angles in Form D. 
The horizontal distances, i. e. the sides of the triangles are now 
calculated (see Forms F and G). Each side should if possible, be 
calculated from two difierent triangles, and the mean of the values 
obtained taken as the true one ; and each station should be fixed by 
at least three lines. When all the sides have been computed they 
should be entered in a table (see Eorm H). 

The last calculation is that of the relative heights of the difi'erent 
stations (see Eorm K.) Erom the vertical angle and the horizon- 
tal distance we obtain the difierence in altitude between the axis 
of the telescope of the theodolite and the signal of the station ob- 
served. To this when cleared of curvature and refraction (see re- 
marks on page 94, also table page 118) must be added the height of 
the theodolite at the time of observation, and when the height of 
the signal observed is deducted, we have the difference of level in 
the ground at these points. 

The triangulation must now be laid down on paper, very carefully, 



157 

by the aid of beam compasses, taking the mean length of the sides 
from the table of distances. When this skeleton triangulation is 
completed, the interior details must be filled in, first by traversing 
the roads or other conspicuous lines with the theodolite, and after- 
wards by sketching with the plane table or compass, as before des- 
cribed in a former part of this work. 



158 



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Form H. (CALCULATION BOOK.) 



Table of Distances. 



Triangles 

calculated 

from. 


Sides. 


Distances 


in feet. 


1 
Remarks. 


Calculated. 


Mean. 




AB, 


1919-568 


1919-568 


Base line. 


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2857-65 




CAD, 
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6493-91 / 


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ECD, 
ECB, 


EC, 
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5601-78 / 


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>> 
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2341-26 ] 

> 

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2340-81 




CDE, 


ED, 

>> 


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3792-68 J 


3792-80 




FGA, 


FA, 


4379-00 ^ 
1 






FCA, 


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1 


4379-02 




FCB, 


» 


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